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

Percentage Accurate: 99.8% → 99.8%

Time: 9.4s

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

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

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 99.9%

   \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
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 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 2: 74.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{z}\\ t_1 := 0.5 \cdot t\_0\\ \mathbf{if}\;t\_0 \leq -4.6 \cdot 10^{-53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 3.75 \cdot 10^{+96}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (sqrt z))) (t_1 (* 0.5 t_0)))
   (if (<= t_0 -4.6e-53) t_1 (if (<= t_0 3.75e+96) (* x 0.5) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = y * sqrt(z);
	double t_1 = 0.5 * t_0;
	double tmp;
	if (t_0 <= -4.6e-53) {
		tmp = t_1;
	} else if (t_0 <= 3.75e+96) {
		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 = 0.5d0 * t_0
    if (t_0 <= (-4.6d-53)) then
        tmp = t_1
    else if (t_0 <= 3.75d+96) 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 = 0.5 * t_0;
	double tmp;
	if (t_0 <= -4.6e-53) {
		tmp = t_1;
	} else if (t_0 <= 3.75e+96) {
		tmp = x * 0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * math.sqrt(z)
	t_1 = 0.5 * t_0
	tmp = 0
	if t_0 <= -4.6e-53:
		tmp = t_1
	elif t_0 <= 3.75e+96:
		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(0.5 * t_0)
	tmp = 0.0
	if (t_0 <= -4.6e-53)
		tmp = t_1;
	elseif (t_0 <= 3.75e+96)
		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 = 0.5 * t_0;
	tmp = 0.0;
	if (t_0 <= -4.6e-53)
		tmp = t_1;
	elseif (t_0 <= 3.75e+96)
		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[(0.5 * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$0, -4.6e-53], t$95$1, If[LessEqual[t$95$0, 3.75e+96], N[(x * 0.5), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (sqrt.f64 z)) < -4.6000000000000003e-53 or 3.7499999999999998e96 < (*.f64 y (sqrt.f64 z))

   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. metadata-eval 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

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

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

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

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

      4. sqrt-lowering-sqrt.f64 87.7

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

   7. Simplified 87.7%

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

   if -4.6000000000000003e-53 < (*.f64 y (sqrt.f64 z)) < 3.7499999999999998e96

   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. +-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 78.5

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

   5. Simplified 78.5%

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

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 78.5

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

   7. Applied egg-rr 78.5%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \sqrt{z} \leq -4.6 \cdot 10^{-53}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{elif}\;y \cdot \sqrt{z} \leq 3.75 \cdot 10^{+96}:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.0% 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 99.9%

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

3. Taylor expanded in x around inf

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

   1. +-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 51.7

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

5. Simplified 51.7%

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

   1. +-rgt-identity N/A

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

   2. *-lowering-*.f64 51.7

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

7. Applied egg-rr 51.7%

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

Reproduce

herbie shell --seed 2024197 
(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)))))