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

Percentage Accurate: 99.8% → 99.8%

Time: 5.7s

Alternatives: 4

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, 1.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return ((sqrt(z) * y) + x) * 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 = ((sqrt(z) * y) + x) * 0.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
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 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 74.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+24}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+25}:\\ \;\;\;\;\left(\sqrt{z} \cdot y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.4e+24)
   (* x 0.5)
   (if (<= x 1.9e+25) (* (* (sqrt z) y) 0.5) (* x 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.4e+24) {
		tmp = x * 0.5;
	} else if (x <= 1.9e+25) {
		tmp = (sqrt(z) * y) * 0.5;
	} else {
		tmp = x * 0.5;
	}
	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 <= (-3.4d+24)) then
        tmp = x * 0.5d0
    else if (x <= 1.9d+25) then
        tmp = (sqrt(z) * y) * 0.5d0
    else
        tmp = x * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.4e+24) {
		tmp = x * 0.5;
	} else if (x <= 1.9e+25) {
		tmp = (Math.sqrt(z) * y) * 0.5;
	} else {
		tmp = x * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.4e+24:
		tmp = x * 0.5
	elif x <= 1.9e+25:
		tmp = (math.sqrt(z) * y) * 0.5
	else:
		tmp = x * 0.5
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.4e+24)
		tmp = Float64(x * 0.5);
	elseif (x <= 1.9e+25)
		tmp = Float64(Float64(sqrt(z) * y) * 0.5);
	else
		tmp = Float64(x * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.4e+24)
		tmp = x * 0.5;
	elseif (x <= 1.9e+25)
		tmp = (sqrt(z) * y) * 0.5;
	else
		tmp = x * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.4e+24], N[(x * 0.5), $MachinePrecision], If[LessEqual[x, 1.9e+25], N[(N[(N[Sqrt[z], $MachinePrecision] * y), $MachinePrecision] * 0.5), $MachinePrecision], N[(x * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.4000000000000001e24 or 1.9e25 < 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 81.9

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

   5. Applied rewrites 81.9%

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

   if -3.4000000000000001e24 < x < 1.9e25

   1. Initial program 99.6%

      \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
   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 76.8

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

   5. Applied rewrites 76.8%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+24}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+25}:\\ \;\;\;\;\left(\sqrt{z} \cdot y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 99.8%

   \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
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 99.8

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

   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 99.7

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

   9. lift-/.f64 N/A

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

   10. metadata-eval 99.7

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

4. Applied rewrites 99.7%

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

Alternative 4: 49.7% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return x * 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 = x * 0.5d0
end function

Java

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

Python

def code(x, y, z):
	return x * 0.5

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
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 52.5

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

5. Applied rewrites 52.5%

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

6. Final simplification 52.5%

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

Reproduce

herbie shell --seed 2024332 
(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)))))