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

Percentage Accurate: 99.8% → 99.8%

Time: 6.3s

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[(0.5 * N[Sqrt[z], $MachinePrecision]), $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. Final simplification 99.8%

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

Alternative 2: 74.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{z}\\ t_1 := t\_0 \cdot 0.5\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+112}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5000000:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (sqrt z))) (t_1 (* t_0 0.5)))
   (if (<= t_0 -2e+112) t_1 (if (<= t_0 5000000.0) (* x 0.5) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = y * sqrt(z);
	double t_1 = t_0 * 0.5;
	double tmp;
	if (t_0 <= -2e+112) {
		tmp = t_1;
	} else if (t_0 <= 5000000.0) {
		tmp = x * 0.5;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = y * sqrt(z)
    t_1 = t_0 * 0.5d0
    if (t_0 <= (-2d+112)) then
        tmp = t_1
    else if (t_0 <= 5000000.0d0) then
        tmp = x * 0.5d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * Math.sqrt(z);
	double t_1 = t_0 * 0.5;
	double tmp;
	if (t_0 <= -2e+112) {
		tmp = t_1;
	} else if (t_0 <= 5000000.0) {
		tmp = x * 0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * math.sqrt(z)
	t_1 = t_0 * 0.5
	tmp = 0
	if t_0 <= -2e+112:
		tmp = t_1
	elif t_0 <= 5000000.0:
		tmp = x * 0.5
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * sqrt(z))
	t_1 = Float64(t_0 * 0.5)
	tmp = 0.0
	if (t_0 <= -2e+112)
		tmp = t_1;
	elseif (t_0 <= 5000000.0)
		tmp = Float64(x * 0.5);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * sqrt(z);
	t_1 = t_0 * 0.5;
	tmp = 0.0;
	if (t_0 <= -2e+112)
		tmp = t_1;
	elseif (t_0 <= 5000000.0)
		tmp = x * 0.5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * 0.5), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+112], t$95$1, If[LessEqual[t$95$0, 5000000.0], N[(x * 0.5), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (sqrt.f64 z)) < -1.9999999999999999e112 or 5e6 < (*.f64 y (sqrt.f64 z))

   1. Initial program 99.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(y \cdot \sqrt{z}\right)} \]
   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 83.2

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

   5. Applied rewrites 83.2%

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

   if -1.9999999999999999e112 < (*.f64 y (sqrt.f64 z)) < 5e6

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \sqrt{z} \leq -2 \cdot 10^{+112}:\\ \;\;\;\;\left(y \cdot \sqrt{z}\right) \cdot 0.5\\ \mathbf{elif}\;y \cdot \sqrt{z} \leq 5000000:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \sqrt{z}\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision] + 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. 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. Final simplification 99.5%

   \[\leadsto \left(y \cdot \sqrt{z} + x\right) \cdot 0.5 \]
6. 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: 50.2% 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 99.5%

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

3. Taylor expanded in y around 0

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

   1. lower-*.f64 51.0

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

5. Applied rewrites 51.0%

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

6. Final simplification 51.0%

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

Reproduce

herbie shell --seed 2024244 
(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)))))