Diagrams.Solve.Polynomial:quartForm from diagrams-solve-0.1, A

Percentage Accurate: 99.9% → 100.0%

Time: 5.9s

Alternatives: 5

Speedup: 1.4×

Specification

Math

\[\begin{array}{l} \\ x - \frac{3}{8} \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- x (* (/ 3.0 8.0) y)))

C

double code(double x, double y) {
	return x - ((3.0 / 8.0) * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x - ((3.0d0 / 8.0d0) * y)
end function

Java

public static double code(double x, double y) {
	return x - ((3.0 / 8.0) * y);
}

Python

def code(x, y):
	return x - ((3.0 / 8.0) * y)

Julia

function code(x, y)
	return Float64(x - Float64(Float64(3.0 / 8.0) * y))
end

MATLAB

function tmp = code(x, y)
	tmp = x - ((3.0 / 8.0) * y);
end

Wolfram

code[x_, y_] := N[(x - N[(N[(3.0 / 8.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \frac{3}{8} \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- x (* (/ 3.0 8.0) y)))

C

double code(double x, double y) {
	return x - ((3.0 / 8.0) * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x - ((3.0d0 / 8.0d0) * y)
end function

Java

public static double code(double x, double y) {
	return x - ((3.0 / 8.0) * y);
}

Python

def code(x, y):
	return x - ((3.0 / 8.0) * y)

Julia

function code(x, y)
	return Float64(x - Float64(Float64(3.0 / 8.0) * y))
end

MATLAB

function tmp = code(x, y)
	tmp = x - ((3.0 / 8.0) * y);
end

Wolfram

code[x_, y_] := N[(x - N[(N[(3.0 / 8.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, -0.375, x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma y -0.375 x))

C

double code(double x, double y) {
	return fma(y, -0.375, x);
}

Julia

function code(x, y)
	return fma(y, -0.375, x)
end

Wolfram

code[x_, y_] := N[(y * -0.375 + x), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x - \frac{3}{8} \cdot y \]
2. Step-by-step derivation

   1. sub-neg 99.8%

      \[\leadsto \color{blue}{x + \left(-\frac{3}{8} \cdot y\right)} \]

   2. distribute-rgt-neg-out 99.8%

      \[\leadsto x + \color{blue}{\frac{3}{8} \cdot \left(-y\right)} \]

   3. +-commutative 99.8%

      \[\leadsto \color{blue}{\frac{3}{8} \cdot \left(-y\right) + x} \]

   4. distribute-rgt-neg-out 99.8%

      \[\leadsto \color{blue}{\left(-\frac{3}{8} \cdot y\right)} + x \]

   5. *-commutative 99.8%

      \[\leadsto \left(-\color{blue}{y \cdot \frac{3}{8}}\right) + x \]

   6. distribute-rgt-neg-in 99.8%

      \[\leadsto \color{blue}{y \cdot \left(-\frac{3}{8}\right)} + x \]

   7. fma-define 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, -\frac{3}{8}, x\right)} \]

   8. metadata-eval 100.0%

      \[\leadsto \mathsf{fma}\left(y, -\color{blue}{0.375}, x\right) \]

   9. metadata-eval 100.0%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{-0.375}, x\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, -0.375, x\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 73.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+90}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5500000:\\ \;\;\;\;y \cdot -0.375\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.8e+90) x (if (<= x 5500000.0) (* y -0.375) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.8e+90) {
		tmp = x;
	} else if (x <= 5500000.0) {
		tmp = y * -0.375;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.8d+90)) then
        tmp = x
    else if (x <= 5500000.0d0) then
        tmp = y * (-0.375d0)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.8e+90) {
		tmp = x;
	} else if (x <= 5500000.0) {
		tmp = y * -0.375;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.8e+90:
		tmp = x
	elif x <= 5500000.0:
		tmp = y * -0.375
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.8e+90)
		tmp = x;
	elseif (x <= 5500000.0)
		tmp = Float64(y * -0.375);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.8e+90)
		tmp = x;
	elseif (x <= 5500000.0)
		tmp = y * -0.375;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.8e+90], x, If[LessEqual[x, 5500000.0], N[(y * -0.375), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.8e90 or 5.5e6 < x

   1. Initial program 99.9%

      \[x - \frac{3}{8} \cdot y \]
   2. Step-by-step derivation

      1. sub-neg 99.9%

         \[\leadsto \color{blue}{x + \left(-\frac{3}{8} \cdot y\right)} \]

      2. *-commutative 99.9%

         \[\leadsto x + \left(-\color{blue}{y \cdot \frac{3}{8}}\right) \]

      3. distribute-rgt-neg-in 99.9%

         \[\leadsto x + \color{blue}{y \cdot \left(-\frac{3}{8}\right)} \]

      4. metadata-eval 99.9%

         \[\leadsto x + y \cdot \left(-\color{blue}{0.375}\right) \]

      5. metadata-eval 99.9%

         \[\leadsto x + y \cdot \color{blue}{-0.375} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x + y \cdot -0.375} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 80.9%

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

   if -2.8e90 < x < 5.5e6

   1. Initial program 99.7%

      \[x - \frac{3}{8} \cdot y \]
   2. Step-by-step derivation

      1. sub-neg 99.7%

         \[\leadsto \color{blue}{x + \left(-\frac{3}{8} \cdot y\right)} \]

      2. *-commutative 99.7%

         \[\leadsto x + \left(-\color{blue}{y \cdot \frac{3}{8}}\right) \]

      3. distribute-rgt-neg-in 99.7%

         \[\leadsto x + \color{blue}{y \cdot \left(-\frac{3}{8}\right)} \]

      4. metadata-eval 99.7%

         \[\leadsto x + y \cdot \left(-\color{blue}{0.375}\right) \]

      5. metadata-eval 99.7%

         \[\leadsto x + y \cdot \color{blue}{-0.375} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x + y \cdot -0.375} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 77.1%

      \[\leadsto \color{blue}{-0.375 \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+90}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5500000:\\ \;\;\;\;y \cdot -0.375\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{-0.375}{\frac{1}{y}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (/ -0.375 (/ 1.0 y))))

C

double code(double x, double y) {
	return x + (-0.375 / (1.0 / y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + ((-0.375d0) / (1.0d0 / y))
end function

Java

public static double code(double x, double y) {
	return x + (-0.375 / (1.0 / y));
}

Python

def code(x, y):
	return x + (-0.375 / (1.0 / y))

Julia

function code(x, y)
	return Float64(x + Float64(-0.375 / Float64(1.0 / y)))
end

MATLAB

function tmp = code(x, y)
	tmp = x + (-0.375 / (1.0 / y));
end

Wolfram

code[x_, y_] := N[(x + N[(-0.375 / N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x - \frac{3}{8} \cdot y \]
2. Step-by-step derivation

   1. sub-neg 99.8%

      \[\leadsto \color{blue}{x + \left(-\frac{3}{8} \cdot y\right)} \]

   2. *-commutative 99.8%

      \[\leadsto x + \left(-\color{blue}{y \cdot \frac{3}{8}}\right) \]

   3. distribute-rgt-neg-in 99.8%

      \[\leadsto x + \color{blue}{y \cdot \left(-\frac{3}{8}\right)} \]

   4. metadata-eval 99.8%

      \[\leadsto x + y \cdot \left(-\color{blue}{0.375}\right) \]

   5. metadata-eval 99.8%

      \[\leadsto x + y \cdot \color{blue}{-0.375} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{x + y \cdot -0.375} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 88.8%

   \[\leadsto \color{blue}{x \cdot \left(1 + -0.375 \cdot \frac{y}{x}\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-in 88.8%

      \[\leadsto \color{blue}{1 \cdot x + \left(-0.375 \cdot \frac{y}{x}\right) \cdot x} \]

   2. *-un-lft-identity 88.8%

      \[\leadsto \color{blue}{x} + \left(-0.375 \cdot \frac{y}{x}\right) \cdot x \]

   3. +-commutative 88.8%

      \[\leadsto \color{blue}{\left(-0.375 \cdot \frac{y}{x}\right) \cdot x + x} \]

   4. *-commutative 88.8%

      \[\leadsto \color{blue}{\left(\frac{y}{x} \cdot -0.375\right)} \cdot x + x \]

   5. associate-*l* 88.8%

      \[\leadsto \color{blue}{\frac{y}{x} \cdot \left(-0.375 \cdot x\right)} + x \]

7. Applied egg-rr 88.8%

   \[\leadsto \color{blue}{\frac{y}{x} \cdot \left(-0.375 \cdot x\right) + x} \]
8. Step-by-step derivation

   1. *-commutative 88.8%

      \[\leadsto \color{blue}{\left(-0.375 \cdot x\right) \cdot \frac{y}{x}} + x \]

   2. clear-num 88.8%

      \[\leadsto \left(-0.375 \cdot x\right) \cdot \color{blue}{\frac{1}{\frac{x}{y}}} + x \]

   3. un-div-inv 89.3%

      \[\leadsto \color{blue}{\frac{-0.375 \cdot x}{\frac{x}{y}}} + x \]

   4. *-commutative 89.3%

      \[\leadsto \frac{\color{blue}{x \cdot -0.375}}{\frac{x}{y}} + x \]

9. Applied egg-rr 89.3%

   \[\leadsto \color{blue}{\frac{x \cdot -0.375}{\frac{x}{y}}} + x \]

11. Applied egg-rr 99.9%

    \[\leadsto \color{blue}{1 \cdot \frac{-0.375}{\frac{1}{y}}} + x \]

12. Final simplification 99.9%

    \[\leadsto x + \frac{-0.375}{\frac{1}{y}} \]
13. Add Preprocessing

Alternative 4: 99.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ x + y \cdot -0.375 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x (* y -0.375)))

C

double code(double x, double y) {
	return x + (y * -0.375);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + (y * (-0.375d0))
end function

Java

public static double code(double x, double y) {
	return x + (y * -0.375);
}

Python

def code(x, y):
	return x + (y * -0.375)

Julia

function code(x, y)
	return Float64(x + Float64(y * -0.375))
end

MATLAB

function tmp = code(x, y)
	tmp = x + (y * -0.375);
end

Wolfram

code[x_, y_] := N[(x + N[(y * -0.375), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[x - \frac{3}{8} \cdot y \]
2. Step-by-step derivation

   1. sub-neg 99.8%

      \[\leadsto \color{blue}{x + \left(-\frac{3}{8} \cdot y\right)} \]

   2. *-commutative 99.8%

      \[\leadsto x + \left(-\color{blue}{y \cdot \frac{3}{8}}\right) \]

   3. distribute-rgt-neg-in 99.8%

      \[\leadsto x + \color{blue}{y \cdot \left(-\frac{3}{8}\right)} \]

   4. metadata-eval 99.8%

      \[\leadsto x + y \cdot \left(-\color{blue}{0.375}\right) \]

   5. metadata-eval 99.8%

      \[\leadsto x + y \cdot \color{blue}{-0.375} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{x + y \cdot -0.375} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 5: 51.2% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

double code(double x, double y) {
	return x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x
end function

Java

public static double code(double x, double y) {
	return x;
}

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

function tmp = code(x, y)
	tmp = x;
end

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 99.8%

   \[x - \frac{3}{8} \cdot y \]
2. Step-by-step derivation

   1. sub-neg 99.8%

      \[\leadsto \color{blue}{x + \left(-\frac{3}{8} \cdot y\right)} \]

   2. *-commutative 99.8%

      \[\leadsto x + \left(-\color{blue}{y \cdot \frac{3}{8}}\right) \]

   3. distribute-rgt-neg-in 99.8%

      \[\leadsto x + \color{blue}{y \cdot \left(-\frac{3}{8}\right)} \]

   4. metadata-eval 99.8%

      \[\leadsto x + y \cdot \left(-\color{blue}{0.375}\right) \]

   5. metadata-eval 99.8%

      \[\leadsto x + y \cdot \color{blue}{-0.375} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{x + y \cdot -0.375} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 51.0%

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

Reproduce

herbie shell --seed 2024085 
(FPCore (x y)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, A"
  :precision binary64
  (- x (* (/ 3.0 8.0) y)))