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

Percentage Accurate: 99.8% → 99.8%

Time: 7.7s

Alternatives: 4

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: 70.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-93}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-138}:\\ \;\;\;\;\frac{0.5 \cdot y}{{z}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5e-93)
   (* 0.5 x)
   (if (<= x 1.95e-138) (/ (* 0.5 y) (pow z -0.5)) (* 0.5 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5e-93) {
		tmp = 0.5 * x;
	} else if (x <= 1.95e-138) {
		tmp = (0.5 * y) / pow(z, -0.5);
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-5d-93)) then
        tmp = 0.5d0 * x
    else if (x <= 1.95d-138) then
        tmp = (0.5d0 * y) / (z ** (-0.5d0))
    else
        tmp = 0.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -5e-93) {
		tmp = 0.5 * x;
	} else if (x <= 1.95e-138) {
		tmp = (0.5 * y) / Math.pow(z, -0.5);
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5e-93:
		tmp = 0.5 * x
	elif x <= 1.95e-138:
		tmp = (0.5 * y) / math.pow(z, -0.5)
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5e-93)
		tmp = Float64(0.5 * x);
	elseif (x <= 1.95e-138)
		tmp = Float64(Float64(0.5 * y) / (z ^ -0.5));
	else
		tmp = Float64(0.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -5e-93)
		tmp = 0.5 * x;
	elseif (x <= 1.95e-138)
		tmp = (0.5 * y) / (z ^ -0.5);
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5e-93], N[(0.5 * x), $MachinePrecision], If[LessEqual[x, 1.95e-138], N[(N[(0.5 * y), $MachinePrecision] / N[Power[z, -0.5], $MachinePrecision]), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.99999999999999994e-93 or 1.95e-138 < x

   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 69.9%

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

   7. Simplified 69.9%

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

   if -4.99999999999999994e-93 < x < 1.95e-138

   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 88.4%

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

   7. Simplified 88.4%

      \[\leadsto \color{blue}{y \cdot \left(0.5 \cdot \sqrt{z}\right)} \]
   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. pow1/2 N/A

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

      5. pow-flip N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. pow1/2 N/A

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

      10. inv-pow N/A

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

      11. associate-*l/ N/A

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

      12. /-lowering-/.f64 N/A

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

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

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

      14. pow1/2 N/A

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

      15. inv-pow N/A

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

      16. pow-prod-up N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

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

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

      20. metadata-eval 88.4%

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

   9. Applied egg-rr 88.4%

      \[\leadsto \color{blue}{\frac{0.5 \cdot y}{{z}^{-0.5}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 70.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-93}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-138}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5e-93)
   (* 0.5 x)
   (if (<= x 1.95e-138) (* y (* 0.5 (sqrt z))) (* 0.5 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5e-93) {
		tmp = 0.5 * x;
	} else if (x <= 1.95e-138) {
		tmp = y * (0.5 * sqrt(z));
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-5d-93)) then
        tmp = 0.5d0 * x
    else if (x <= 1.95d-138) then
        tmp = y * (0.5d0 * sqrt(z))
    else
        tmp = 0.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -5e-93) {
		tmp = 0.5 * x;
	} else if (x <= 1.95e-138) {
		tmp = y * (0.5 * Math.sqrt(z));
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5e-93:
		tmp = 0.5 * x
	elif x <= 1.95e-138:
		tmp = y * (0.5 * math.sqrt(z))
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5e-93)
		tmp = Float64(0.5 * x);
	elseif (x <= 1.95e-138)
		tmp = Float64(y * Float64(0.5 * sqrt(z)));
	else
		tmp = Float64(0.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -5e-93)
		tmp = 0.5 * x;
	elseif (x <= 1.95e-138)
		tmp = y * (0.5 * sqrt(z));
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5e-93], N[(0.5 * x), $MachinePrecision], If[LessEqual[x, 1.95e-138], N[(y * N[(0.5 * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.99999999999999994e-93 or 1.95e-138 < x

   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 69.9%

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

   7. Simplified 69.9%

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

   if -4.99999999999999994e-93 < x < 1.95e-138

   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 88.4%

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

   7. Simplified 88.4%

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

Alternative 4: 50.2% accurate, 36.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-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 53.3%

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

7. Simplified 53.3%

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

Reproduce

herbie shell --seed 2024158 
(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)))))