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

Percentage Accurate: 99.8% → 99.8%

Time: 6.2s

Alternatives: 5

Speedup: 1.6×

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, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x + y \cdot \sqrt{z}\right)} \]

   3. +-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(y \cdot \sqrt{z} + x\right)} \]

   4. distribute-lft-in N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(y \cdot \sqrt{z}\right) + \frac{1}{2} \cdot x} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(y \cdot \sqrt{z}\right)} + \frac{1}{2} \cdot x \]

   6. *-commutative N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{z} \cdot y\right)} + \frac{1}{2} \cdot x \]

   7. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sqrt{z}\right) \cdot y} + \frac{1}{2} \cdot x \]

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 99.8

       \[\leadsto \mathsf{fma}\left(\sqrt{z} \cdot 0.5, y, \color{blue}{x \cdot \frac{1}{2}}\right) \]

   15. lift-/.f64 N/A

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

   16. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 74.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8500 \lor \neg \left(x \leq 55\right):\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{z} \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -8500.0) (not (<= x 55.0))) (* 0.5 x) (* 0.5 (* (sqrt z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8500.0) || !(x <= 55.0)) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * (sqrt(z) * y);
	}
	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 <= (-8500.0d0)) .or. (.not. (x <= 55.0d0))) then
        tmp = 0.5d0 * x
    else
        tmp = 0.5d0 * (sqrt(z) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8500.0) || !(x <= 55.0)) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * (Math.sqrt(z) * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -8500.0) or not (x <= 55.0):
		tmp = 0.5 * x
	else:
		tmp = 0.5 * (math.sqrt(z) * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -8500.0) || !(x <= 55.0))
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(0.5 * Float64(sqrt(z) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -8500.0) || ~((x <= 55.0)))
		tmp = 0.5 * x;
	else
		tmp = 0.5 * (sqrt(z) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -8500.0], N[Not[LessEqual[x, 55.0]], $MachinePrecision]], N[(0.5 * x), $MachinePrecision], N[(0.5 * N[(N[Sqrt[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8500 or 55 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
   4. Step-by-step derivation

      1. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

   if -8500 < x < 55

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(y \cdot \sqrt{z}\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{z} \cdot y\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\sqrt{z} \cdot y\right)} \]

      4. lower-sqrt.f64 76.4

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

   5. Applied rewrites 76.4%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8500 \lor \neg \left(x \leq 55\right):\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{z} \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.5× 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. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot \left(x + y \cdot \sqrt{z}\right) \]

   2. metadata-eval 99.5

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

4. Applied rewrites 99.5%

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

Alternative 4: 99.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(x + y \cdot \sqrt{z}\right) \cdot \frac{1}{2}} \]

   3. lower-*.f64 99.5

      \[\leadsto \color{blue}{\left(x + y \cdot \sqrt{z}\right) \cdot \frac{1}{2}} \]

   4. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(x + y \cdot \sqrt{z}\right)} \cdot \frac{1}{2} \]

   5. +-commutative N/A

      \[\leadsto \color{blue}{\left(y \cdot \sqrt{z} + x\right)} \cdot \frac{1}{2} \]

   6. lift-*.f64 N/A

      \[\leadsto \left(\color{blue}{y \cdot \sqrt{z}} + x\right) \cdot \frac{1}{2} \]

   7. *-commutative N/A

      \[\leadsto \left(\color{blue}{\sqrt{z} \cdot y} + x\right) \cdot \frac{1}{2} \]

   8. lower-fma.f64 99.5

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

   9. lift-/.f64 N/A

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

   10. metadata-eval 99.5

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

4. Applied rewrites 99.5%

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

Alternative 5: 50.8% accurate, 5.8× 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. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
4. Step-by-step derivation

   1. lower-*.f64 53.6

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

5. Applied rewrites 53.6%

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

Reproduce

herbie shell --seed 2024338 
(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)))))