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

Percentage Accurate: 99.8% → 99.8%

Time: 5.8s

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, 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 99.8%

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

   1. metadata-eval 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 72.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+95}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.5e+95)
   (* 0.5 x)
   (if (<= x 3.6e-21) (* 0.5 (* y (sqrt z))) (* 0.5 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.5e+95) {
		tmp = 0.5 * x;
	} else if (x <= 3.6e-21) {
		tmp = 0.5 * (y * 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 <= (-3.5d+95)) then
        tmp = 0.5d0 * x
    else if (x <= 3.6d-21) then
        tmp = 0.5d0 * (y * 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 <= -3.5e+95) {
		tmp = 0.5 * x;
	} else if (x <= 3.6e-21) {
		tmp = 0.5 * (y * Math.sqrt(z));
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.5e+95:
		tmp = 0.5 * x
	elif x <= 3.6e-21:
		tmp = 0.5 * (y * math.sqrt(z))
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.5e+95)
		tmp = Float64(0.5 * x);
	elseif (x <= 3.6e-21)
		tmp = Float64(0.5 * Float64(y * 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 <= -3.5e+95)
		tmp = 0.5 * x;
	elseif (x <= 3.6e-21)
		tmp = 0.5 * (y * sqrt(z));
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.5e+95], N[(0.5 * x), $MachinePrecision], If[LessEqual[x, 3.6e-21], N[(0.5 * N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5e95 or 3.59999999999999989e-21 < x

   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. Taylor expanded in x around inf 84.4%

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

   if -3.5e95 < x < 3.59999999999999989e-21

   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. Taylor expanded in x around 0 78.9%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+95}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-21}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * (z ** 0.5d0)))
end function

Java

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

Python

def code(x, y, z):
	return 0.5 * (x + (y * math.pow(z, 0.5)))

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(0.5 * N[(x + N[(y * N[Power[z, 0.5], $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. add-sqr-sqrt 47.8%

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

   2. sqrt-unprod 59.4%

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

   3. pow1/2 59.4%

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

   4. *-commutative 59.4%

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

   5. *-commutative 59.4%

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

   6. swap-sqr 55.7%

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

   7. add-sqr-sqrt 55.7%

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

5. Applied egg-rr 55.7%

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

   1. unpow1/2 55.7%

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

7. Simplified 55.7%

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

   1. sqrt-prod 60.0%

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

   2. pow1/2 60.0%

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

   3. metadata-eval 60.0%

      \[\leadsto 0.5 \cdot \left(x + {z}^{\color{blue}{\left(0.25 + 0.25\right)}} \cdot \sqrt{y \cdot y}\right) \]

   4. pow-prod-up 59.9%

      \[\leadsto 0.5 \cdot \left(x + \color{blue}{\left({z}^{0.25} \cdot {z}^{0.25}\right)} \cdot \sqrt{y \cdot y}\right) \]

   5. sqrt-prod 45.0%

      \[\leadsto 0.5 \cdot \left(x + \left({z}^{0.25} \cdot {z}^{0.25}\right) \cdot \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)}\right) \]

   6. add-sqr-sqrt 99.5%

      \[\leadsto 0.5 \cdot \left(x + \left({z}^{0.25} \cdot {z}^{0.25}\right) \cdot \color{blue}{y}\right) \]

   7. associate-*r* 99.5%

      \[\leadsto 0.5 \cdot \left(x + \color{blue}{{z}^{0.25} \cdot \left({z}^{0.25} \cdot y\right)}\right) \]

9. Applied egg-rr 99.5%

   \[\leadsto 0.5 \cdot \left(x + \color{blue}{{z}^{0.25} \cdot \left({z}^{0.25} \cdot y\right)}\right) \]
10. Step-by-step derivation

    1. associate-*r* 99.5%

       \[\leadsto 0.5 \cdot \left(x + \color{blue}{\left({z}^{0.25} \cdot {z}^{0.25}\right) \cdot y}\right) \]

    2. *-commutative 99.5%

       \[\leadsto 0.5 \cdot \left(x + \color{blue}{y \cdot \left({z}^{0.25} \cdot {z}^{0.25}\right)}\right) \]

    3. pow-sqr 99.8%

       \[\leadsto 0.5 \cdot \left(x + y \cdot \color{blue}{{z}^{\left(2 \cdot 0.25\right)}}\right) \]

    4. metadata-eval 99.8%

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

11. Simplified 99.8%

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

12. Final simplification 99.8%

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

Alternative 4: 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 5: 49.8% 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 46.3%

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

5. Final simplification 46.3%

   \[\leadsto 0.5 \cdot x \]

Reproduce

herbie shell --seed 2023229 
(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)))))