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

Percentage Accurate: 99.8% → 99.8%

Time: 8.2s

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 99.5%

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

Alternative 2: 73.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-42}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-39}:\\ \;\;\;\;\sqrt{z} \cdot \left(0.5 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left|x\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2e-42)
   (* 0.5 x)
   (if (<= x 1.45e-39) (* (sqrt z) (* 0.5 y)) (* 0.5 (fabs x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2e-42) {
		tmp = 0.5 * x;
	} else if (x <= 1.45e-39) {
		tmp = sqrt(z) * (0.5 * y);
	} else {
		tmp = 0.5 * fabs(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 <= (-2d-42)) then
        tmp = 0.5d0 * x
    else if (x <= 1.45d-39) then
        tmp = sqrt(z) * (0.5d0 * y)
    else
        tmp = 0.5d0 * abs(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2e-42) {
		tmp = 0.5 * x;
	} else if (x <= 1.45e-39) {
		tmp = Math.sqrt(z) * (0.5 * y);
	} else {
		tmp = 0.5 * Math.abs(x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2e-42:
		tmp = 0.5 * x
	elif x <= 1.45e-39:
		tmp = math.sqrt(z) * (0.5 * y)
	else:
		tmp = 0.5 * math.fabs(x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2e-42)
		tmp = Float64(0.5 * x);
	elseif (x <= 1.45e-39)
		tmp = Float64(sqrt(z) * Float64(0.5 * y));
	else
		tmp = Float64(0.5 * abs(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2e-42)
		tmp = 0.5 * x;
	elseif (x <= 1.45e-39)
		tmp = sqrt(z) * (0.5 * y);
	else
		tmp = 0.5 * abs(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2e-42], N[(0.5 * x), $MachinePrecision], If[LessEqual[x, 1.45e-39], N[(N[Sqrt[z], $MachinePrecision] * N[(0.5 * y), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Abs[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.00000000000000008e-42

   1. Initial program 98.7%

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

      1. metadata-eval 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in x around inf 72.5%

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

   if -2.00000000000000008e-42 < x < 1.44999999999999994e-39

   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. Add Preprocessing

   5. Taylor expanded in x around 0 81.2%

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

      1. associate-*r* 81.2%

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

      2. *-commutative 81.2%

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

      3. *-commutative 81.2%

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

   7. Simplified 81.2%

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

   if 1.44999999999999994e-39 < 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 z around inf 86.8%

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

   6. Taylor expanded in y around 0 64.7%

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

      1. associate-*r/ 67.5%

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

      2. clear-num 67.3%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{z}{z \cdot x}}} \]

      3. *-commutative 67.3%

         \[\leadsto 0.5 \cdot \frac{1}{\frac{z}{\color{blue}{x \cdot z}}} \]

   8. Applied egg-rr 67.3%

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

      1. *-commutative 67.3%

         \[\leadsto 0.5 \cdot \frac{1}{\frac{z}{\color{blue}{z \cdot x}}} \]

      2. associate-/r* 77.2%

         \[\leadsto 0.5 \cdot \frac{1}{\color{blue}{\frac{\frac{z}{z}}{x}}} \]

      3. *-inverses 77.2%

         \[\leadsto 0.5 \cdot \frac{1}{\frac{\color{blue}{1}}{x}} \]

   10. Simplified 77.2%

       \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{1}{x}}} \]
   11. Step-by-step derivation

       1. remove-double-div 77.4%

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

       2. add-sqr-sqrt 76.9%

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

       3. sqrt-unprod 48.3%

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

       4. pow2 48.3%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{x}^{2}}} \]

   12. Applied egg-rr 48.3%

       \[\leadsto 0.5 \cdot \color{blue}{\sqrt{{x}^{2}}} \]
   13. Step-by-step derivation

       1. unpow2 48.3%

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

       2. rem-sqrt-square 77.4%

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

   14. Simplified 77.4%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-42}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-39}:\\ \;\;\;\;\sqrt{z} \cdot \left(0.5 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left|x\right|\\ \end{array} \]
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 99.5%

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

Alternative 4: 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. Add Preprocessing

5. Taylor expanded in x around inf 47.9%

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

Reproduce

herbie shell --seed 2024172 
(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)))))