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

Percentage Accurate: 99.8% → 99.8%

Time: 9.0s

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, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[(y * N[Sqrt[z], $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. *-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 99.8

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

4. Applied egg-rr 99.8%

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

Alternative 2: 75.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{z}\\ t_1 := y \cdot \left(\sqrt{z} \cdot 0.5\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+34}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (sqrt z))) (t_1 (* y (* (sqrt z) 0.5))))
   (if (<= t_0 -2e+21) t_1 (if (<= t_0 2e+34) (* x 0.5) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = y * sqrt(z);
	double t_1 = y * (sqrt(z) * 0.5);
	double tmp;
	if (t_0 <= -2e+21) {
		tmp = t_1;
	} else if (t_0 <= 2e+34) {
		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 = y * (sqrt(z) * 0.5d0)
    if (t_0 <= (-2d+21)) then
        tmp = t_1
    else if (t_0 <= 2d+34) 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 = y * (Math.sqrt(z) * 0.5);
	double tmp;
	if (t_0 <= -2e+21) {
		tmp = t_1;
	} else if (t_0 <= 2e+34) {
		tmp = x * 0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * math.sqrt(z)
	t_1 = y * (math.sqrt(z) * 0.5)
	tmp = 0
	if t_0 <= -2e+21:
		tmp = t_1
	elif t_0 <= 2e+34:
		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(y * Float64(sqrt(z) * 0.5))
	tmp = 0.0
	if (t_0 <= -2e+21)
		tmp = t_1;
	elseif (t_0 <= 2e+34)
		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 = y * (sqrt(z) * 0.5);
	tmp = 0.0;
	if (t_0 <= -2e+21)
		tmp = t_1;
	elseif (t_0 <= 2e+34)
		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[(y * N[(N[Sqrt[z], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+21], t$95$1, If[LessEqual[t$95$0, 2e+34], N[(x * 0.5), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (sqrt.f64 z)) < -2e21 or 1.99999999999999989e34 < (*.f64 y (sqrt.f64 z))

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

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

      7. sqrt-lowering-sqrt.f64 87.0

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

   5. Simplified 87.0%

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

   if -2e21 < (*.f64 y (sqrt.f64 z)) < 1.99999999999999989e34

   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. *-commutative N/A

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

      2. *-lowering-*.f64 77.3

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

   5. Simplified 77.3%

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

Alternative 3: 50.3% 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. *-commutative N/A

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

   2. *-lowering-*.f64 46.9

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

5. Simplified 46.9%

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

Alternative 4: 2.3% 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. *-commutative N/A

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

   2. *-lowering-*.f64 46.9

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

5. Simplified 46.9%

   \[\leadsto \color{blue}{x \cdot 0.5} \]
6. Step-by-step derivation

   1. remove-double-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \cdot \frac{1}{2} \]

   2. neg-sub0 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(0 - x\right)}\right)\right) \cdot \frac{1}{2} \]

   3. flip3-- N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}}\right)\right) \cdot \frac{1}{2} \]

   4. metadata-eval N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{0} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right)\right) \cdot \frac{1}{2} \]

   5. neg-sub0 N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left({x}^{3}\right)}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right)\right) \cdot \frac{1}{2} \]

   6. cube-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{3}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right)\right) \cdot \frac{1}{2} \]

   7. sqr-pow N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right)\right) \cdot \frac{1}{2} \]

   8. unpow-prod-down N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right)\right) \cdot \frac{1}{2} \]

   9. sqr-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{{\color{blue}{\left(x \cdot x\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right)\right) \cdot \frac{1}{2} \]

   10. unpow-prod-down N/A

       \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{{x}^{\left(\frac{3}{2}\right)} \cdot {x}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right)\right) \cdot \frac{1}{2} \]

   11. sqr-pow N/A

       \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{{x}^{3}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right)\right) \cdot \frac{1}{2} \]

   12. distribute-neg-frac N/A

       \[\leadsto \color{blue}{\frac{\mathsf{neg}\left({x}^{3}\right)}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}} \cdot \frac{1}{2} \]

   13. neg-sub0 N/A

       \[\leadsto \frac{\color{blue}{0 - {x}^{3}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \cdot \frac{1}{2} \]

   14. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{{0}^{3}} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \cdot \frac{1}{2} \]

   15. flip3-- N/A

       \[\leadsto \color{blue}{\left(0 - x\right)} \cdot \frac{1}{2} \]

   16. neg-sub0 N/A

       \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \frac{1}{2} \]

   17. distribute-lft-neg-in N/A

       \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \frac{1}{2}\right)} \]

   18. distribute-rgt-neg-in N/A

       \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

   19. metadata-eval N/A

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

   20. metadata-eval N/A

       \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \frac{1}{2}\right)} \]

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

       \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \frac{1}{2}\right)} \]

7. Applied egg-rr 2.5%

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

Reproduce

herbie shell --seed 2024199 
(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)))))