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

Percentage Accurate: 99.8% → 99.8%

Time: 28.7s

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, 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 99.9%

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

   1. metadata-eval 99.9%

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

   2. +-commutative 99.9%

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

   3. fma-define 99.9%

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

3. Simplified 99.9%

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

Alternative 2: 73.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -70000000000000:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{z} \cdot \left(0.5 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left|0.5 \cdot x\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -70000000000000.0)
   (* 0.5 x)
   (if (<= x 5.4e-68) (* (sqrt z) (* 0.5 y)) (fabs (* 0.5 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -70000000000000.0) {
		tmp = 0.5 * x;
	} else if (x <= 5.4e-68) {
		tmp = sqrt(z) * (0.5 * y);
	} else {
		tmp = fabs((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 <= (-70000000000000.0d0)) then
        tmp = 0.5d0 * x
    else if (x <= 5.4d-68) then
        tmp = sqrt(z) * (0.5d0 * y)
    else
        tmp = abs((0.5d0 * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -70000000000000.0) {
		tmp = 0.5 * x;
	} else if (x <= 5.4e-68) {
		tmp = Math.sqrt(z) * (0.5 * y);
	} else {
		tmp = Math.abs((0.5 * x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -70000000000000.0:
		tmp = 0.5 * x
	elif x <= 5.4e-68:
		tmp = math.sqrt(z) * (0.5 * y)
	else:
		tmp = math.fabs((0.5 * x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -70000000000000.0)
		tmp = Float64(0.5 * x);
	elseif (x <= 5.4e-68)
		tmp = Float64(sqrt(z) * Float64(0.5 * y));
	else
		tmp = abs(Float64(0.5 * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -70000000000000.0)
		tmp = 0.5 * x;
	elseif (x <= 5.4e-68)
		tmp = sqrt(z) * (0.5 * y);
	else
		tmp = abs((0.5 * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -70000000000000.0], N[(0.5 * x), $MachinePrecision], If[LessEqual[x, 5.4e-68], N[(N[Sqrt[z], $MachinePrecision] * N[(0.5 * y), $MachinePrecision]), $MachinePrecision], N[Abs[N[(0.5 * x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7e13

   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. Add Preprocessing

   5. Taylor expanded in x around inf 81.4%

      \[\leadsto \color{blue}{0.5 \cdot x} \]
   6. Step-by-step derivation

      1. *-commutative 81.4%

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

   7. Simplified 81.4%

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

   if -7e13 < x < 5.4000000000000003e-68

   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. Add Preprocessing

   5. Taylor expanded in x around 0 82.2%

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

      1. associate-*r* 82.2%

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

      2. *-commutative 82.2%

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

      3. *-commutative 82.2%

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

   7. Simplified 82.2%

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

   if 5.4000000000000003e-68 < 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. Add Preprocessing

   5. Taylor expanded in x around inf 67.9%

      \[\leadsto \color{blue}{0.5 \cdot x} \]
   6. Step-by-step derivation

      1. *-commutative 67.9%

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

   7. Simplified 67.9%

      \[\leadsto \color{blue}{x \cdot 0.5} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 67.3%

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

      2. sqrt-unprod 45.9%

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

      3. *-commutative 45.9%

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

      4. *-commutative 45.9%

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

      5. swap-sqr 45.9%

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

      6. metadata-eval 45.9%

         \[\leadsto \sqrt{\color{blue}{0.25} \cdot \left(x \cdot x\right)} \]

      7. pow2 45.9%

         \[\leadsto \sqrt{0.25 \cdot \color{blue}{{x}^{2}}} \]

   9. Applied egg-rr 45.9%

      \[\leadsto \color{blue}{\sqrt{0.25 \cdot {x}^{2}}} \]
   10. Step-by-step derivation

       1. *-commutative 45.9%

          \[\leadsto \sqrt{\color{blue}{{x}^{2} \cdot 0.25}} \]

       2. unpow2 45.9%

          \[\leadsto \sqrt{\color{blue}{\left(x \cdot x\right)} \cdot 0.25} \]

       3. metadata-eval 45.9%

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

       4. swap-sqr 45.9%

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

       5. rem-sqrt-square 67.9%

          \[\leadsto \color{blue}{\left|x \cdot 0.5\right|} \]

   11. Simplified 67.9%

       \[\leadsto \color{blue}{\left|x \cdot 0.5\right|} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -70000000000000:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{z} \cdot \left(0.5 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left|0.5 \cdot x\right|\\ \end{array} \]
5. Add Preprocessing

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 99.9%

   \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
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. Add Preprocessing
5. Add Preprocessing

Alternative 4: 51.3% 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 99.9%

   \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
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. Add Preprocessing

5. Taylor expanded in x around inf 51.1%

   \[\leadsto \color{blue}{0.5 \cdot x} \]
6. Step-by-step derivation

   1. *-commutative 51.1%

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

7. Simplified 51.1%

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

8. Final simplification 51.1%

   \[\leadsto 0.5 \cdot x \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024116 
(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)))))