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

Percentage Accurate: 99.8% → 99.6%

Time: 9.9s

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: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({z}^{0.25}, {z}^{0.25} \cdot \left(0.5 \cdot y\right), 0.5 \cdot x\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (fma (pow z 0.25) (* (pow z 0.25) (* 0.5 y)) (* 0.5 x)))

C

double code(double x, double y, double z) {
	return fma(pow(z, 0.25), (pow(z, 0.25) * (0.5 * y)), (0.5 * x));
}

Julia

function code(x, y, z)
	return fma((z ^ 0.25), Float64((z ^ 0.25) * Float64(0.5 * y)), Float64(0.5 * x))
end

Wolfram

code[x_, y_, z_] := N[(N[Power[z, 0.25], $MachinePrecision] * N[(N[Power[z, 0.25], $MachinePrecision] * N[(0.5 * y), $MachinePrecision]), $MachinePrecision] + N[(0.5 * x), $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. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. pow1/2 N/A

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

   6. metadata-eval N/A

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

   7. sqr-pow N/A

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

   8. associate-*l* N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

4. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left({z}^{0.25}, {z}^{0.25} \cdot \left(0.5 \cdot y\right), 0.5 \cdot x\right)} \]
5. Add Preprocessing

Alternative 2: 73.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1350:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 6.1 \cdot 10^{-28}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1350.0)
   (* 0.5 x)
   (if (<= x 6.1e-28) (* (* 0.5 y) (sqrt z)) (* 0.5 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1350.0) {
		tmp = 0.5 * x;
	} else if (x <= 6.1e-28) {
		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 (x <= (-1350.0d0)) then
        tmp = 0.5d0 * x
    else if (x <= 6.1d-28) 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 (x <= -1350.0) {
		tmp = 0.5 * x;
	} else if (x <= 6.1e-28) {
		tmp = (0.5 * y) * Math.sqrt(z);
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1350.0:
		tmp = 0.5 * x
	elif x <= 6.1e-28:
		tmp = (0.5 * y) * math.sqrt(z)
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1350.0)
		tmp = Float64(0.5 * x);
	elseif (x <= 6.1e-28)
		tmp = Float64(Float64(0.5 * 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 (x <= -1350.0)
		tmp = 0.5 * x;
	elseif (x <= 6.1e-28)
		tmp = (0.5 * y) * sqrt(z);
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1350.0], N[(0.5 * x), $MachinePrecision], If[LessEqual[x, 6.1e-28], N[(N[(0.5 * y), $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1350 or 6.1e-28 < x

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot x + 0} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{2}} + 0 \]

      3. accelerator-lowering-fma.f64 77.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, 0\right)} \]

   5. Simplified 77.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{2}} \]

      2. *-lowering-*.f64 77.8

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

   7. Applied egg-rr 77.8%

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

   if -1350 < x < 6.1e-28

   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. +-rgt-identity N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(y \cdot \sqrt{z}\right) + 0} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot \sqrt{z}\right) \cdot \frac{1}{2}} + 0 \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{y \cdot \left(\sqrt{z} \cdot \frac{1}{2}\right)} + 0 \]

      4. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{1}{2} \cdot \sqrt{z}\right)} + 0 \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{1}{2} \cdot \sqrt{z}, 0\right)} \]

      6. +-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{0 + \frac{1}{2} \cdot \sqrt{z}}, 0\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{2} \cdot \sqrt{z} + 0}, 0\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\sqrt{z} \cdot \frac{1}{2}} + 0, 0\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\sqrt{z}, \frac{1}{2}, 0\right)}, 0\right) \]

      10. sqrt-lowering-sqrt.f64 78.3

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(\color{blue}{\sqrt{z}}, 0.5, 0\right), 0\right) \]

   5. Simplified 78.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(\sqrt{z}, 0.5, 0\right), 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{y \cdot \left(\sqrt{z} \cdot \frac{1}{2} + 0\right)} \]

      2. +-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 78.3

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

   7. Applied egg-rr 78.3%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1350:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{elif}\;x \leq 6.1 \cdot 10^{-28}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 99.5%

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

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

   3. +-commutative N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

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

   6. metadata-eval 99.5

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

4. Applied egg-rr 99.5%

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

5. Final simplification 99.5%

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

Alternative 4: 51.2% 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. +-rgt-identity N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + 0} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{x \cdot \frac{1}{2}} + 0 \]

   3. accelerator-lowering-fma.f64 50.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, 0\right)} \]

5. Simplified 50.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{x \cdot \frac{1}{2}} \]

   2. *-lowering-*.f64 50.4

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

7. Applied egg-rr 50.4%

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

8. Final simplification 50.4%

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

Reproduce

herbie shell --seed 2024196 
(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)))))