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

Percentage Accurate: 99.8% → 99.8%

Time: 6.7s

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, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (* 0.5 (+ (/ y (pow z -0.5)) x)))

C

double code(double x, double y, double z) {
	return 0.5 * ((y / pow(z, -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 * ((y / (z ** (-0.5d0))) + x)
end function

Java

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

Python

def code(x, y, z):
	return 0.5 * ((y / math.pow(z, -0.5)) + x)

Julia

function code(x, y, z)
	return Float64(0.5 * Float64(Float64(y / (z ^ -0.5)) + x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.5 * ((y / (z ^ -0.5)) + x);
end

Wolfram

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

Derivation

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%

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

   2. +-commutative 99.8%

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

   3. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in z around inf 80.2%

   \[\leadsto 0.5 \cdot \color{blue}{\left(z \cdot \left(y \cdot \sqrt{\frac{1}{z}} + \frac{x}{z}\right)\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-in 80.2%

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

   2. associate-*l* 85.1%

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

   3. fma-define 85.1%

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

   4. *-commutative 85.1%

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

   5. associate-*l/ 84.7%

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

   6. associate-/l* 99.8%

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

   7. *-inverses 99.8%

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

   8. *-rgt-identity 99.8%

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

7. Simplified 99.8%

   \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(y, z \cdot \sqrt{\frac{1}{z}}, x\right)} \]
8. Step-by-step derivation

   1. fma-undefine 99.8%

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

   2. *-commutative 99.8%

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

   3. sqrt-div 99.7%

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

   4. metadata-eval 99.7%

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

   5. un-div-inv 99.7%

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

9. Applied egg-rr 99.7%

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

    1. *-commutative 99.7%

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

    2. clear-num 99.7%

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

    3. un-div-inv 99.8%

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

    4. pow1/2 99.8%

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

    5. pow1 99.8%

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

    6. pow-div 99.8%

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

    7. metadata-eval 99.8%

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

11. Applied egg-rr 99.8%

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

12. Final simplification 99.8%

    \[\leadsto 0.5 \cdot \left(\frac{y}{{z}^{-0.5}} + x\right) \]
13. Add Preprocessing

Alternative 2: 73.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -112000 \lor \neg \left(y \leq 7.2 \cdot 10^{-6}\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -112000.0) (not (<= y 7.2e-6)))
   (* 0.5 (* y (sqrt z)))
   (* 0.5 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -112000.0) || !(y <= 7.2e-6)) {
		tmp = 0.5 * (y * sqrt(z));
	} else {
		tmp = 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 ((y <= (-112000.0d0)) .or. (.not. (y <= 7.2d-6))) then
        tmp = 0.5d0 * (y * sqrt(z))
    else
        tmp = 0.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -112000.0) || !(y <= 7.2e-6)) {
		tmp = 0.5 * (y * Math.sqrt(z));
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -112000.0) or not (y <= 7.2e-6):
		tmp = 0.5 * (y * math.sqrt(z))
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -112000.0) || !(y <= 7.2e-6))
		tmp = Float64(0.5 * Float64(y * sqrt(z)));
	else
		tmp = Float64(0.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -112000.0) || ~((y <= 7.2e-6)))
		tmp = 0.5 * (y * sqrt(z));
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -112000.0], N[Not[LessEqual[y, 7.2e-6]], $MachinePrecision]], N[(0.5 * N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -112000 or 7.19999999999999967e-6 < y

   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%

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

      2. +-commutative 99.7%

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

      3. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around inf 76.9%

      \[\leadsto 0.5 \cdot \color{blue}{\left(z \cdot \left(y \cdot \sqrt{\frac{1}{z}} + \frac{x}{z}\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-rgt-in 76.9%

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

      2. associate-*l* 85.9%

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

      3. fma-define 85.9%

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

      4. *-commutative 85.9%

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

      5. associate-*l/ 89.9%

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

      6. associate-/l* 99.6%

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

      7. *-inverses 99.6%

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

      8. *-rgt-identity 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in y around inf 77.5%

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

      1. *-commutative 77.5%

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

   10. Simplified 77.5%

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

   if -112000 < y < 7.19999999999999967e-6

   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. Taylor expanded in y around 0 79.4%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -112000 \lor \neg \left(y \leq 7.2 \cdot 10^{-6}\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]
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 99.8%

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

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

Alternative 4: 50.7% 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 99.8%

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

   1. metadata-eval 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in y around 0 53.5%

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

6. Final simplification 53.5%

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

Reproduce

herbie shell --seed 2024081 
(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)))))