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

Percentage Accurate: 99.8% → 99.8%

Time: 10.9s

Alternatives: 6

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))

C

double code(double x, double y, double z) {
	return (1.0 / 2.0) * (x + (y * sqrt(z)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (1.0d0 / 2.0d0) * (x + (y * sqrt(z)))
end function

Java

public static double code(double x, double y, double z) {
	return (1.0 / 2.0) * (x + (y * Math.sqrt(z)));
}

Python

def code(x, y, z):
	return (1.0 / 2.0) * (x + (y * math.sqrt(z)))

Julia

function code(x, y, z)
	return Float64(Float64(1.0 / 2.0) * Float64(x + Float64(y * sqrt(z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (1.0 / 2.0) * (x + (y * sqrt(z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / 2.0), $MachinePrecision] * N[(x + N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))

C

double code(double x, double y, double z) {
	return (1.0 / 2.0) * (x + (y * sqrt(z)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (1.0d0 / 2.0d0) * (x + (y * sqrt(z)))
end function

Java

public static double code(double x, double y, double z) {
	return (1.0 / 2.0) * (x + (y * Math.sqrt(z)));
}

Python

def code(x, y, z):
	return (1.0 / 2.0) * (x + (y * math.sqrt(z)))

Julia

function code(x, y, z)
	return Float64(Float64(1.0 / 2.0) * Float64(x + Float64(y * sqrt(z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (1.0 / 2.0) * (x + (y * sqrt(z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / 2.0), $MachinePrecision] * N[(x + N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. metadata-eval 99.5%

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

   2. +-commutative 99.5%

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

   3. fma-def 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 2: 72.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+47}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq -12200 \lor \neg \left(x \leq -9.8 \cdot 10^{-24}\right) \land x \leq 7.5 \cdot 10^{+72}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.8e+47)
   (* 0.5 x)
   (if (or (<= x -12200.0) (and (not (<= x -9.8e-24)) (<= x 7.5e+72)))
     (* y (* 0.5 (sqrt z)))
     (* 0.5 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.8e+47) {
		tmp = 0.5 * x;
	} else if ((x <= -12200.0) || (!(x <= -9.8e-24) && (x <= 7.5e+72))) {
		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 <= (-3.8d+47)) then
        tmp = 0.5d0 * x
    else if ((x <= (-12200.0d0)) .or. (.not. (x <= (-9.8d-24))) .and. (x <= 7.5d+72)) 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 <= -3.8e+47) {
		tmp = 0.5 * x;
	} else if ((x <= -12200.0) || (!(x <= -9.8e-24) && (x <= 7.5e+72))) {
		tmp = y * (0.5 * Math.sqrt(z));
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.8e+47:
		tmp = 0.5 * x
	elif (x <= -12200.0) or (not (x <= -9.8e-24) and (x <= 7.5e+72)):
		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 <= -3.8e+47)
		tmp = Float64(0.5 * x);
	elseif ((x <= -12200.0) || (!(x <= -9.8e-24) && (x <= 7.5e+72)))
		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 <= -3.8e+47)
		tmp = 0.5 * x;
	elseif ((x <= -12200.0) || (~((x <= -9.8e-24)) && (x <= 7.5e+72)))
		tmp = y * (0.5 * sqrt(z));
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.8e+47], N[(0.5 * x), $MachinePrecision], If[Or[LessEqual[x, -12200.0], And[N[Not[LessEqual[x, -9.8e-24]], $MachinePrecision], LessEqual[x, 7.5e+72]]], 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 < -3.8000000000000003e47 or -12200 < x < -9.8000000000000002e-24 or 7.50000000000000027e72 < 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. Taylor expanded in x around inf 81.7%

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

   if -3.8000000000000003e47 < x < -12200 or -9.8000000000000002e-24 < x < 7.50000000000000027e72

   1. Initial program 99.2%

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

      1. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 77.1%

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

      1. associate-*r* 77.7%

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

      2. *-commutative 77.7%

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

      3. associate-*r* 77.7%

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

   6. Simplified 77.7%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+47}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq -12200 \lor \neg \left(x \leq -9.8 \cdot 10^{-24}\right) \land x \leq 7.5 \cdot 10^{+72}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 3: 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 99.5%

   \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
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. Final simplification 99.5%

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

Alternative 4: 51.6% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{+234}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(z \cdot \left(y \cdot \frac{y}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.4e+234) (* 0.5 x) (* 0.5 (* z (* y (/ y x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.4e+234) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * (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 (y <= 4.4d+234) then
        tmp = 0.5d0 * x
    else
        tmp = 0.5d0 * (z * (y * (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.4e+234) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * (z * (y * (y / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.4e+234:
		tmp = 0.5 * x
	else:
		tmp = 0.5 * (z * (y * (y / x)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.4e+234)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(0.5 * Float64(z * Float64(y * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.4e+234)
		tmp = 0.5 * x;
	else
		tmp = 0.5 * (z * (y * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.4e+234], N[(0.5 * x), $MachinePrecision], N[(0.5 * N[(z * N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.40000000000000015e234

   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 inf 53.2%

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

   if 4.40000000000000015e234 < y

   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-+ 14.0%

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

      2. associate-*r/ 14.0%

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

      3. *-commutative 14.0%

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

      4. *-commutative 14.0%

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

      5. swap-sqr 1.0%

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

      6. add-sqr-sqrt 1.0%

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

   5. Applied egg-rr 1.0%

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

      1. associate-/l* 1.0%

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

   7. Simplified 1.0%

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

   8. Taylor expanded in x around 0 1.4%

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

      1. mul-1-neg 1.4%

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

      2. unpow2 1.4%

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

      3. associate-*r* 13.9%

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

      4. distribute-rgt-neg-out 13.9%

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

      5. distribute-lft-neg-in 13.9%

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

   10. Simplified 13.9%

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

   11. Taylor expanded in x around inf 27.1%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{{y}^{2} \cdot z}{x}} \]
   12. Step-by-step derivation

       1. unpow2 27.1%

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

       2. associate-*r* 27.2%

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

       3. associate-/l* 27.2%

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

   13. Simplified 27.2%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{y}{\frac{x}{y \cdot z}}} \]
   14. Step-by-step derivation

       1. associate-*r/ 27.2%

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

       2. frac-2neg 27.2%

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

       3. metadata-eval 27.2%

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

       4. distribute-lft-neg-in 27.2%

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

       5. *-commutative 27.2%

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

       6. distribute-rgt-neg-in 27.2%

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

       7. metadata-eval 27.2%

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

       8. distribute-neg-frac 27.2%

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

       9. add-sqr-sqrt 27.1%

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

       10. sqrt-unprod 54.6%

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

       11. sqr-neg 54.6%

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

       12. sqrt-unprod 41.1%

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

       13. add-sqr-sqrt 41.6%

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

   15. Applied egg-rr 41.6%

       \[\leadsto \color{blue}{\frac{-y \cdot -0.5}{\frac{x}{y \cdot z}}} \]
   16. Step-by-step derivation

       1. distribute-rgt-neg-in 41.6%

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

       2. metadata-eval 41.6%

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

       3. *-commutative 41.6%

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

       4. associate-*r/ 41.6%

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

       5. associate-/l* 41.5%

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

       6. associate-*r* 41.3%

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

       7. associate-*l/ 41.3%

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

       8. *-commutative 41.3%

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

       9. associate-*r/ 41.3%

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

   17. Simplified 41.3%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{+234}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(z \cdot \left(y \cdot \frac{y}{x}\right)\right)\\ \end{array} \]

Alternative 5: 51.6% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{+236}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \frac{0.5}{x}\right) \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.9e+236) (* 0.5 x) (* (* y (/ 0.5 x)) (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.9e+236) {
		tmp = 0.5 * x;
	} else {
		tmp = (y * (0.5 / x)) * (y * 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) :: tmp
    if (y <= 1.9d+236) then
        tmp = 0.5d0 * x
    else
        tmp = (y * (0.5d0 / x)) * (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.9e+236) {
		tmp = 0.5 * x;
	} else {
		tmp = (y * (0.5 / x)) * (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.9e+236:
		tmp = 0.5 * x
	else:
		tmp = (y * (0.5 / x)) * (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.9e+236)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(Float64(y * Float64(0.5 / x)) * Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.9e+236)
		tmp = 0.5 * x;
	else
		tmp = (y * (0.5 / x)) * (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.9e+236], N[(0.5 * x), $MachinePrecision], N[(N[(y * N[(0.5 / x), $MachinePrecision]), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.89999999999999993e236

   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 inf 53.2%

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

   if 1.89999999999999993e236 < y

   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-+ 14.0%

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

      2. associate-*r/ 14.0%

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

      3. *-commutative 14.0%

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

      4. *-commutative 14.0%

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

      5. swap-sqr 1.0%

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

      6. add-sqr-sqrt 1.0%

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

   5. Applied egg-rr 1.0%

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

      1. associate-/l* 1.0%

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

   7. Simplified 1.0%

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

   8. Taylor expanded in x around 0 1.4%

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

      1. mul-1-neg 1.4%

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

      2. unpow2 1.4%

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

      3. associate-*r* 13.9%

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

      4. distribute-rgt-neg-out 13.9%

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

      5. distribute-lft-neg-in 13.9%

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

   10. Simplified 13.9%

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

   11. Taylor expanded in x around inf 27.1%

       \[\leadsto \frac{0.5}{\color{blue}{-1 \cdot \frac{x}{{y}^{2} \cdot z}}} \]
   12. Step-by-step derivation

       1. associate-*r/ 27.1%

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

       2. neg-mul-1 27.1%

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

       3. unpow2 27.1%

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

       4. associate-*r* 27.2%

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

   13. Simplified 27.2%

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

       1. associate-/r/ 27.2%

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

       2. associate-*r* 27.2%

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

       3. add-sqr-sqrt 27.1%

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

       4. sqrt-unprod 54.6%

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

       5. sqr-neg 54.6%

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

       6. sqrt-unprod 41.1%

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

       7. add-sqr-sqrt 41.6%

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

   15. Applied egg-rr 41.6%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{+236}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \frac{0.5}{x}\right) \cdot \left(y \cdot z\right)\\ \end{array} \]

Alternative 6: 50.9% 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 99.5%

   \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
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 inf 50.6%

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

5. Final simplification 50.6%

   \[\leadsto 0.5 \cdot x \]

Reproduce

herbie shell --seed 2023275 
(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)))))