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

Percentage Accurate: 100.0% → 100.0%

Time: 3.4s

Alternatives: 5

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{2} - \frac{z}{8} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (/ (* x y) 2.0) (/ z 8.0)))

C

double code(double x, double y, double z) {
	return ((x * y) / 2.0) - (z / 8.0);
}

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 * y) / 2.0d0) - (z / 8.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return ((x * y) / 2.0) - (z / 8.0);
}

Python

def code(x, y, z):
	return ((x * y) / 2.0) - (z / 8.0)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x * y) / 2.0) - Float64(z / 8.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x * y) / 2.0) - (z / 8.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x * y), $MachinePrecision] / 2.0), $MachinePrecision] - N[(z / 8.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{2} - \frac{z}{8} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (/ (* x y) 2.0) (/ z 8.0)))

C

double code(double x, double y, double z) {
	return ((x * y) / 2.0) - (z / 8.0);
}

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 * y) / 2.0d0) - (z / 8.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return ((x * y) / 2.0) - (z / 8.0);
}

Python

def code(x, y, z):
	return ((x * y) / 2.0) - (z / 8.0)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x * y) / 2.0) - Float64(z / 8.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x * y) / 2.0) - (z / 8.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x * y), $MachinePrecision] / 2.0), $MachinePrecision] - N[(z / 8.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{x}{2}, y, -0.125 \cdot z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (/ x 2.0) y (* -0.125 z)))

C

double code(double x, double y, double z) {
	return fma((x / 2.0), y, (-0.125 * z));
}

Julia

function code(x, y, z)
	return fma(Float64(x / 2.0), y, Float64(-0.125 * z))
end

Wolfram

code[x_, y_, z_] := N[(N[(x / 2.0), $MachinePrecision] * y + N[(-0.125 * z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. associate-*l/ 100.0%

      \[\leadsto \color{blue}{\frac{x}{2} \cdot y} - \frac{z}{8} \]

   2. fma-neg 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{2}, y, -\frac{z}{8}\right)} \]

   3. distribute-frac-neg 100.0%

      \[\leadsto \mathsf{fma}\left(\frac{x}{2}, y, \color{blue}{\frac{-z}{8}}\right) \]

   4. neg-mul-1 100.0%

      \[\leadsto \mathsf{fma}\left(\frac{x}{2}, y, \frac{\color{blue}{-1 \cdot z}}{8}\right) \]

   5. associate-/l* 99.9%

      \[\leadsto \mathsf{fma}\left(\frac{x}{2}, y, \color{blue}{\frac{-1}{\frac{8}{z}}}\right) \]

   6. associate-/r/ 100.0%

      \[\leadsto \mathsf{fma}\left(\frac{x}{2}, y, \color{blue}{\frac{-1}{8} \cdot z}\right) \]

   7. metadata-eval 100.0%

      \[\leadsto \mathsf{fma}\left(\frac{x}{2}, y, \color{blue}{-0.125} \cdot z\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{2}, y, -0.125 \cdot z\right)} \]
4. Add Preprocessing

5. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(\frac{x}{2}, y, -0.125 \cdot z\right) \]
6. Add Preprocessing

Alternative 2: 69.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -130000 \lor \neg \left(x \leq 7.2 \cdot 10^{+52}\right):\\ \;\;\;\;x \cdot \left(y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -130000.0) (not (<= x 7.2e+52))) (* x (* y 0.5)) (* -0.125 z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -130000.0) || !(x <= 7.2e+52)) {
		tmp = x * (y * 0.5);
	} else {
		tmp = -0.125 * z;
	}
	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 <= (-130000.0d0)) .or. (.not. (x <= 7.2d+52))) then
        tmp = x * (y * 0.5d0)
    else
        tmp = (-0.125d0) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -130000.0) || !(x <= 7.2e+52)) {
		tmp = x * (y * 0.5);
	} else {
		tmp = -0.125 * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -130000.0) or not (x <= 7.2e+52):
		tmp = x * (y * 0.5)
	else:
		tmp = -0.125 * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -130000.0) || !(x <= 7.2e+52))
		tmp = Float64(x * Float64(y * 0.5));
	else
		tmp = Float64(-0.125 * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -130000.0) || ~((x <= 7.2e+52)))
		tmp = x * (y * 0.5);
	else
		tmp = -0.125 * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -130000.0], N[Not[LessEqual[x, 7.2e+52]], $MachinePrecision]], N[(x * N[(y * 0.5), $MachinePrecision]), $MachinePrecision], N[(-0.125 * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3e5 or 7.2e52 < x

   1. Initial program 100.0%

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

      1. associate-/l* 99.8%

         \[\leadsto \color{blue}{\frac{x}{\frac{2}{y}}} - \frac{z}{8} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x}{\frac{2}{y}} - \frac{z}{8}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/l* 100.0%

         \[\leadsto \color{blue}{\frac{x \cdot y}{2}} - \frac{z}{8} \]

      2. frac-2neg 100.0%

         \[\leadsto \color{blue}{\frac{-x \cdot y}{-2}} - \frac{z}{8} \]

      3. clear-num 99.9%

         \[\leadsto \frac{-x \cdot y}{-2} - \color{blue}{\frac{1}{\frac{8}{z}}} \]

      4. frac-sub 86.6%

         \[\leadsto \color{blue}{\frac{\left(-x \cdot y\right) \cdot \frac{8}{z} - \left(-2\right) \cdot 1}{\left(-2\right) \cdot \frac{8}{z}}} \]

      5. metadata-eval 86.6%

         \[\leadsto \frac{\left(-x \cdot y\right) \cdot \frac{8}{z} - \color{blue}{-2} \cdot 1}{\left(-2\right) \cdot \frac{8}{z}} \]

      6. metadata-eval 86.6%

         \[\leadsto \frac{\left(-x \cdot y\right) \cdot \frac{8}{z} - \color{blue}{-2}}{\left(-2\right) \cdot \frac{8}{z}} \]

      7. metadata-eval 86.6%

         \[\leadsto \frac{\left(-x \cdot y\right) \cdot \frac{8}{z} - \color{blue}{\left(-2\right)}}{\left(-2\right) \cdot \frac{8}{z}} \]

      8. fma-neg 86.6%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-x \cdot y, \frac{8}{z}, -\left(-2\right)\right)}}{\left(-2\right) \cdot \frac{8}{z}} \]

      9. *-commutative 86.6%

         \[\leadsto \frac{\mathsf{fma}\left(-\color{blue}{y \cdot x}, \frac{8}{z}, -\left(-2\right)\right)}{\left(-2\right) \cdot \frac{8}{z}} \]

      10. distribute-rgt-neg-in 86.6%

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{y \cdot \left(-x\right)}, \frac{8}{z}, -\left(-2\right)\right)}{\left(-2\right) \cdot \frac{8}{z}} \]

      11. metadata-eval 86.6%

          \[\leadsto \frac{\mathsf{fma}\left(y \cdot \left(-x\right), \frac{8}{z}, -\color{blue}{-2}\right)}{\left(-2\right) \cdot \frac{8}{z}} \]

      12. metadata-eval 86.6%

          \[\leadsto \frac{\mathsf{fma}\left(y \cdot \left(-x\right), \frac{8}{z}, \color{blue}{2}\right)}{\left(-2\right) \cdot \frac{8}{z}} \]

      13. metadata-eval 86.6%

          \[\leadsto \frac{\mathsf{fma}\left(y \cdot \left(-x\right), \frac{8}{z}, 2\right)}{\color{blue}{-2} \cdot \frac{8}{z}} \]

   6. Applied egg-rr 86.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y \cdot \left(-x\right), \frac{8}{z}, 2\right)}{-2 \cdot \frac{8}{z}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 86.6%

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot \left(-x\right), \frac{8}{z}, 2\right)}{\color{blue}{\frac{-2 \cdot 8}{z}}} \]

      2. metadata-eval 86.6%

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot \left(-x\right), \frac{8}{z}, 2\right)}{\frac{\color{blue}{-16}}{z}} \]

      3. associate-/r/ 86.6%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y \cdot \left(-x\right), \frac{8}{z}, 2\right)}{-16} \cdot z} \]

      4. fma-udef 86.6%

         \[\leadsto \frac{\color{blue}{\left(y \cdot \left(-x\right)\right) \cdot \frac{8}{z} + 2}}{-16} \cdot z \]

      5. +-commutative 86.6%

         \[\leadsto \frac{\color{blue}{2 + \left(y \cdot \left(-x\right)\right) \cdot \frac{8}{z}}}{-16} \cdot z \]

      6. associate-*r/ 86.7%

         \[\leadsto \frac{2 + \color{blue}{\frac{\left(y \cdot \left(-x\right)\right) \cdot 8}{z}}}{-16} \cdot z \]

      7. distribute-rgt-neg-out 86.7%

         \[\leadsto \frac{2 + \frac{\color{blue}{\left(-y \cdot x\right)} \cdot 8}{z}}{-16} \cdot z \]

      8. *-commutative 86.7%

         \[\leadsto \frac{2 + \frac{\left(-\color{blue}{x \cdot y}\right) \cdot 8}{z}}{-16} \cdot z \]

      9. distribute-lft-neg-out 86.7%

         \[\leadsto \frac{2 + \frac{\color{blue}{-\left(x \cdot y\right) \cdot 8}}{z}}{-16} \cdot z \]

      10. distribute-rgt-neg-in 86.7%

          \[\leadsto \frac{2 + \frac{\color{blue}{\left(x \cdot y\right) \cdot \left(-8\right)}}{z}}{-16} \cdot z \]

      11. metadata-eval 86.7%

          \[\leadsto \frac{2 + \frac{\left(x \cdot y\right) \cdot \color{blue}{-8}}{z}}{-16} \cdot z \]

      12. associate-*l* 86.7%

          \[\leadsto \frac{2 + \frac{\color{blue}{x \cdot \left(y \cdot -8\right)}}{z}}{-16} \cdot z \]

   8. Simplified 86.7%

      \[\leadsto \color{blue}{\frac{2 + \frac{x \cdot \left(y \cdot -8\right)}{z}}{-16} \cdot z} \]

   9. Taylor expanded in x around inf 75.2%

      \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot y\right)} \]
   10. Step-by-step derivation

       1. *-commutative 75.2%

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

       2. associate-*l* 75.2%

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

   11. Simplified 75.2%

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

   if -1.3e5 < x < 7.2e52

   1. Initial program 100.0%

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

      1. associate-/l* 99.8%

         \[\leadsto \color{blue}{\frac{x}{\frac{2}{y}}} - \frac{z}{8} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x}{\frac{2}{y}} - \frac{z}{8}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 67.9%

      \[\leadsto \color{blue}{-0.125 \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -130000 \lor \neg \left(x \leq 7.2 \cdot 10^{+52}\right):\\ \;\;\;\;x \cdot \left(y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\frac{2}{y}} - \frac{z}{8} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (/ x (/ 2.0 y)) (/ z 8.0)))

C

double code(double x, double y, double z) {
	return (x / (2.0 / y)) - (z / 8.0);
}

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 / (2.0d0 / y)) - (z / 8.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (x / (2.0 / y)) - (z / 8.0);
}

Python

def code(x, y, z):
	return (x / (2.0 / y)) - (z / 8.0)

Julia

function code(x, y, z)
	return Float64(Float64(x / Float64(2.0 / y)) - Float64(z / 8.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x / (2.0 / y)) - (z / 8.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(x / N[(2.0 / y), $MachinePrecision]), $MachinePrecision] - N[(z / 8.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. associate-/l* 99.8%

      \[\leadsto \color{blue}{\frac{x}{\frac{2}{y}}} - \frac{z}{8} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\frac{x}{\frac{2}{y}} - \frac{z}{8}} \]
4. Add Preprocessing

5. Final simplification 99.8%

   \[\leadsto \frac{x}{\frac{2}{y}} - \frac{z}{8} \]
6. Add Preprocessing

Alternative 4: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{2} - \frac{z}{8} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (/ (* x y) 2.0) (/ z 8.0)))

C

double code(double x, double y, double z) {
	return ((x * y) / 2.0) - (z / 8.0);
}

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 * y) / 2.0d0) - (z / 8.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return ((x * y) / 2.0) - (z / 8.0);
}

Python

def code(x, y, z):
	return ((x * y) / 2.0) - (z / 8.0)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x * y) / 2.0) - Float64(z / 8.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x * y) / 2.0) - (z / 8.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x * y), $MachinePrecision] / 2.0), $MachinePrecision] - N[(z / 8.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{x \cdot y}{2} - \frac{z}{8} \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \frac{x \cdot y}{2} - \frac{z}{8} \]
4. Add Preprocessing

Alternative 5: 50.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ -0.125 \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* -0.125 z))

C

double code(double x, double y, double z) {
	return -0.125 * 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 = (-0.125d0) * z
end function

Java

public static double code(double x, double y, double z) {
	return -0.125 * z;
}

Python

def code(x, y, z):
	return -0.125 * z

Julia

function code(x, y, z)
	return Float64(-0.125 * z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = -0.125 * z;
end

Wolfram

code[x_, y_, z_] := N[(-0.125 * z), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. associate-/l* 99.8%

      \[\leadsto \color{blue}{\frac{x}{\frac{2}{y}}} - \frac{z}{8} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\frac{x}{\frac{2}{y}} - \frac{z}{8}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 49.0%

   \[\leadsto \color{blue}{-0.125 \cdot z} \]

6. Final simplification 49.0%

   \[\leadsto -0.125 \cdot z \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024026 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, D"
  :precision binary64
  (- (/ (* x y) 2.0) (/ z 8.0)))