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

Percentage Accurate: 99.8% → 99.8%

Time: 4.0s

Alternatives: 3

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, 1.6× 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. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. *-rgt-identity N/A

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

   2. *-inverses N/A

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

   3. associate-/l* N/A

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

   4. associate-*l/ N/A

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

   5. associate-*r/ N/A

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

   6. associate-*l/ N/A

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

   7. associate-*l* N/A

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

   8. distribute-lft-out N/A

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

   9. *-lft-identity N/A

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

   10. distribute-rgt-in N/A

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

   11. lower-*.f64 N/A

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

   12. distribute-rgt-in N/A

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

   13. *-commutative N/A

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

   14. associate-*l/ N/A

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

   15. associate-*r/ N/A

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

   16. associate-*l/ N/A

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

   17. *-inverses N/A

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

   18. distribute-lft-out N/A

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

5. Simplified 99.8%

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

Alternative 2: 73.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(0.5 \cdot \sqrt{z}\right)\\ \mathbf{if}\;y \leq -2.5 \cdot 10^{-29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.85 \cdot 10^{-74}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (* 0.5 (sqrt z)))))
   (if (<= y -2.5e-29) t_0 (if (<= y 2.85e-74) (* 0.5 x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * (0.5 * sqrt(z));
	double tmp;
	if (y <= -2.5e-29) {
		tmp = t_0;
	} else if (y <= 2.85e-74) {
		tmp = 0.5 * x;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = y * (0.5d0 * sqrt(z))
    if (y <= (-2.5d-29)) then
        tmp = t_0
    else if (y <= 2.85d-74) then
        tmp = 0.5d0 * x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (0.5 * Math.sqrt(z));
	double tmp;
	if (y <= -2.5e-29) {
		tmp = t_0;
	} else if (y <= 2.85e-74) {
		tmp = 0.5 * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (0.5 * math.sqrt(z))
	tmp = 0
	if y <= -2.5e-29:
		tmp = t_0
	elif y <= 2.85e-74:
		tmp = 0.5 * x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(0.5 * sqrt(z)))
	tmp = 0.0
	if (y <= -2.5e-29)
		tmp = t_0;
	elseif (y <= 2.85e-74)
		tmp = Float64(0.5 * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (0.5 * sqrt(z));
	tmp = 0.0;
	if (y <= -2.5e-29)
		tmp = t_0;
	elseif (y <= 2.85e-74)
		tmp = 0.5 * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(0.5 * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.5e-29], t$95$0, If[LessEqual[y, 2.85e-74], N[(0.5 * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.49999999999999993e-29 or 2.85000000000000012e-74 < y

   1. Initial program 99.7%

      \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
   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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sqrt.f64 77.1

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

   5. Simplified 77.1%

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

   if -2.49999999999999993e-29 < y < 2.85000000000000012e-74

   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 83.0

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

   5. Simplified 83.0%

      \[\leadsto \color{blue}{0.5 \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-29}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \sqrt{z}\right)\\ \mathbf{elif}\;y \leq 2.85 \cdot 10^{-74}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.0% 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 99.8%

   \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
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 48.7

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

5. Simplified 48.7%

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

Reproduce

herbie shell --seed 2024215 
(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)))))