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

Percentage Accurate: 99.8% → 99.8%

Time: 9.0s

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: 72.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5 \cdot y}{{z}^{-0.5}}\\ \mathbf{if}\;y \leq -9.6 \cdot 10^{+62}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-12}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+73}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+134}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \sqrt{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 0.5 y) (pow z -0.5))))
   (if (<= y -9.6e+62)
     t_0
     (if (<= y 3.1e-12)
       (* 0.5 x)
       (if (<= y 5.4e+73)
         t_0
         (if (<= y 1.25e+134) (* 0.5 x) (* y (* 0.5 (sqrt z)))))))))

C

double code(double x, double y, double z) {
	double t_0 = (0.5 * y) / pow(z, -0.5);
	double tmp;
	if (y <= -9.6e+62) {
		tmp = t_0;
	} else if (y <= 3.1e-12) {
		tmp = 0.5 * x;
	} else if (y <= 5.4e+73) {
		tmp = t_0;
	} else if (y <= 1.25e+134) {
		tmp = 0.5 * x;
	} else {
		tmp = y * (0.5 * sqrt(z));
	}
	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 <= (-9.6d+62)) then
        tmp = t_0
    else if (y <= 3.1d-12) then
        tmp = 0.5d0 * x
    else if (y <= 5.4d+73) then
        tmp = t_0
    else if (y <= 1.25d+134) then
        tmp = 0.5d0 * x
    else
        tmp = y * (0.5d0 * sqrt(z))
    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 <= -9.6e+62) {
		tmp = t_0;
	} else if (y <= 3.1e-12) {
		tmp = 0.5 * x;
	} else if (y <= 5.4e+73) {
		tmp = t_0;
	} else if (y <= 1.25e+134) {
		tmp = 0.5 * x;
	} else {
		tmp = y * (0.5 * Math.sqrt(z));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (0.5 * y) / math.pow(z, -0.5)
	tmp = 0
	if y <= -9.6e+62:
		tmp = t_0
	elif y <= 3.1e-12:
		tmp = 0.5 * x
	elif y <= 5.4e+73:
		tmp = t_0
	elif y <= 1.25e+134:
		tmp = 0.5 * x
	else:
		tmp = y * (0.5 * math.sqrt(z))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(0.5 * y) / (z ^ -0.5))
	tmp = 0.0
	if (y <= -9.6e+62)
		tmp = t_0;
	elseif (y <= 3.1e-12)
		tmp = Float64(0.5 * x);
	elseif (y <= 5.4e+73)
		tmp = t_0;
	elseif (y <= 1.25e+134)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(y * Float64(0.5 * sqrt(z)));
	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 <= -9.6e+62)
		tmp = t_0;
	elseif (y <= 3.1e-12)
		tmp = 0.5 * x;
	elseif (y <= 5.4e+73)
		tmp = t_0;
	elseif (y <= 1.25e+134)
		tmp = 0.5 * x;
	else
		tmp = y * (0.5 * sqrt(z));
	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, -9.6e+62], t$95$0, If[LessEqual[y, 3.1e-12], N[(0.5 * x), $MachinePrecision], If[LessEqual[y, 5.4e+73], t$95$0, If[LessEqual[y, 1.25e+134], N[(0.5 * x), $MachinePrecision], N[(y * N[(0.5 * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.6e62 or 3.1000000000000001e-12 < y < 5.3999999999999998e73

   1. Initial program 99.6%

      \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
   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 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(x + y \cdot \sqrt{z}\right)} \]
   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 82.1%

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

   7. Simplified 82.1%

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. associate-/r/ N/A

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

      4. metadata-eval N/A

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

      5. sqrt-div N/A

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

      6. associate-*l/ N/A

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

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

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

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

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

      9. sqrt-div N/A

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

      10. metadata-eval N/A

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

      11. pow1/2 N/A

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

      12. pow-flip N/A

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

      13. pow-lowering-pow.f64 N/A

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

      14. metadata-eval 82.3%

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

   9. Applied egg-rr 82.3%

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

   if -9.6e62 < y < 3.1000000000000001e-12 or 5.3999999999999998e73 < y < 1.24999999999999995e134

   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 82.2%

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

   7. Simplified 82.2%

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

   if 1.24999999999999995e134 < y

   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 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 65.2%

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

   7. Simplified 65.2%

      \[\leadsto \color{blue}{y \cdot \left(0.5 \cdot \sqrt{z}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 72.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(0.5 \cdot \sqrt{z}\right)\\ \mathbf{if}\;y \leq -3.1 \cdot 10^{+62}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-10}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+73}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+134}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (* 0.5 (sqrt z)))))
   (if (<= y -3.1e+62)
     t_0
     (if (<= y 3.1e-10)
       (* 0.5 x)
       (if (<= y 1.4e+73) t_0 (if (<= y 2.1e+134) (* 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 <= -3.1e+62) {
		tmp = t_0;
	} else if (y <= 3.1e-10) {
		tmp = 0.5 * x;
	} else if (y <= 1.4e+73) {
		tmp = t_0;
	} else if (y <= 2.1e+134) {
		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 <= (-3.1d+62)) then
        tmp = t_0
    else if (y <= 3.1d-10) then
        tmp = 0.5d0 * x
    else if (y <= 1.4d+73) then
        tmp = t_0
    else if (y <= 2.1d+134) 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 <= -3.1e+62) {
		tmp = t_0;
	} else if (y <= 3.1e-10) {
		tmp = 0.5 * x;
	} else if (y <= 1.4e+73) {
		tmp = t_0;
	} else if (y <= 2.1e+134) {
		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 <= -3.1e+62:
		tmp = t_0
	elif y <= 3.1e-10:
		tmp = 0.5 * x
	elif y <= 1.4e+73:
		tmp = t_0
	elif y <= 2.1e+134:
		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 <= -3.1e+62)
		tmp = t_0;
	elseif (y <= 3.1e-10)
		tmp = Float64(0.5 * x);
	elseif (y <= 1.4e+73)
		tmp = t_0;
	elseif (y <= 2.1e+134)
		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 <= -3.1e+62)
		tmp = t_0;
	elseif (y <= 3.1e-10)
		tmp = 0.5 * x;
	elseif (y <= 1.4e+73)
		tmp = t_0;
	elseif (y <= 2.1e+134)
		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, -3.1e+62], t$95$0, If[LessEqual[y, 3.1e-10], N[(0.5 * x), $MachinePrecision], If[LessEqual[y, 1.4e+73], t$95$0, If[LessEqual[y, 2.1e+134], N[(0.5 * x), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.10000000000000014e62 or 3.10000000000000015e-10 < y < 1.40000000000000004e73 or 2.1000000000000001e134 < 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 76.1%

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

   7. Simplified 76.1%

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

   if -3.10000000000000014e62 < y < 3.10000000000000015e-10 or 1.40000000000000004e73 < y < 2.1000000000000001e134

   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 82.2%

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

   7. Simplified 82.2%

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

Alternative 4: 52.8% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.3 \cdot 10^{+114}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{0.5 \cdot x}{z} + -0.5 \cdot \frac{y \cdot y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 3.3e+114)
   (* 0.5 x)
   (* z (+ (/ (* 0.5 x) z) (* -0.5 (/ (* y y) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 3.3e+114) {
		tmp = 0.5 * x;
	} else {
		tmp = z * (((0.5 * x) / z) + (-0.5 * ((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 (z <= 3.3d+114) then
        tmp = 0.5d0 * x
    else
        tmp = z * (((0.5d0 * x) / z) + ((-0.5d0) * ((y * y) / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 3.3e+114) {
		tmp = 0.5 * x;
	} else {
		tmp = z * (((0.5 * x) / z) + (-0.5 * ((y * y) / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 3.3e+114:
		tmp = 0.5 * x
	else:
		tmp = z * (((0.5 * x) / z) + (-0.5 * ((y * y) / x)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 3.3e+114)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(z * Float64(Float64(Float64(0.5 * x) / z) + Float64(-0.5 * Float64(Float64(y * y) / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 3.3e+114)
		tmp = 0.5 * x;
	else
		tmp = z * (((0.5 * x) / z) + (-0.5 * ((y * y) / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 3.3e+114], N[(0.5 * x), $MachinePrecision], N[(z * N[(N[(N[(0.5 * x), $MachinePrecision] / z), $MachinePrecision] + N[(-0.5 * N[(N[(y * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.3000000000000001e114

   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 61.6%

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

   7. Simplified 61.6%

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

   if 3.3000000000000001e114 < z

   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. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

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

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

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

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

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

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

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

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

      9. rem-square-sqrt N/A

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

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

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

      11. rem-square-sqrt N/A

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

      12. *-commutative N/A

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

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

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

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

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

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

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

   6. Applied egg-rr 42.3%

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

   7. Taylor expanded in x around inf

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

   9. Simplified 26.5%

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

   10. Taylor expanded in z around inf

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

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

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

       2. +-commutative N/A

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

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

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

       4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 52.9%

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

   12. Simplified 52.9%

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

Alternative 5: 50.6% 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 57.0%

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

7. Simplified 57.0%

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

Reproduce

herbie shell --seed 2024149 
(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)))))