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

Percentage Accurate: 100.0% → 100.0%

Time: 6.5s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. associate-*r/ 100.0%

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

   2. *-commutative 100.0%

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

   3. *-commutative 100.0%

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

   4. fma-neg 100.0%

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

   5. distribute-neg-frac 100.0%

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

   6. neg-mul-1 100.0%

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

   7. associate-/l* 99.9%

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

   8. associate-/r/ 100.0%

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

   9. *-commutative 100.0%

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

   10. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 67.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-179} \lor \neg \left(y \leq 460000000\right):\\ \;\;\;\;y \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot -0.125\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.2e-179) (not (<= y 460000000.0)))
   (* y (* x 0.5))
   (* z -0.125)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.2e-179) || !(y <= 460000000.0)) {
		tmp = y * (x * 0.5);
	} else {
		tmp = z * -0.125;
	}
	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 ((y <= (-2.2d-179)) .or. (.not. (y <= 460000000.0d0))) then
        tmp = y * (x * 0.5d0)
    else
        tmp = z * (-0.125d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.2e-179) || !(y <= 460000000.0)) {
		tmp = y * (x * 0.5);
	} else {
		tmp = z * -0.125;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.2e-179) or not (y <= 460000000.0):
		tmp = y * (x * 0.5)
	else:
		tmp = z * -0.125
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.2e-179) || !(y <= 460000000.0))
		tmp = Float64(y * Float64(x * 0.5));
	else
		tmp = Float64(z * -0.125);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.2e-179) || ~((y <= 460000000.0)))
		tmp = y * (x * 0.5);
	else
		tmp = z * -0.125;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.2e-179], N[Not[LessEqual[y, 460000000.0]], $MachinePrecision]], N[(y * N[(x * 0.5), $MachinePrecision]), $MachinePrecision], N[(z * -0.125), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.20000000000000005e-179 or 4.6e8 < y

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

      2. frac-2neg 99.9%

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

      3. clear-num 99.8%

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

      4. frac-sub 91.2%

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

      5. metadata-eval 91.2%

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

      6. metadata-eval 91.2%

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

      7. metadata-eval 91.2%

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

      8. fma-neg 91.2%

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

      9. distribute-rgt-neg-in 91.2%

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

      10. metadata-eval 91.2%

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

      11. metadata-eval 91.2%

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

      12. metadata-eval 91.2%

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

   5. Applied egg-rr 91.2%

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

      1. associate-*r/ 91.2%

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

      2. metadata-eval 91.2%

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

      3. associate-/r/ 91.3%

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

   7. Simplified 91.4%

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

   8. Taylor expanded in y around inf 61.1%

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

      1. *-commutative 61.1%

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

      2. associate-*l* 61.1%

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

   10. Simplified 61.1%

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

   if -2.20000000000000005e-179 < y < 4.6e8

   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. Taylor expanded in x around 0 64.0%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-179} \lor \neg \left(y \leq 460000000\right):\\ \;\;\;\;y \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot -0.125\\ \end{array} \]

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. Final simplification 99.8%

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

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. Final simplification 100.0%

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

Alternative 5: 50.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = z * (-0.125d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(z * -0.125), $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. Taylor expanded in x around 0 48.9%

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

5. Final simplification 48.9%

   \[\leadsto z \cdot -0.125 \]

Reproduce

herbie shell --seed 2023200 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, D"
  :precision binary64
  (- (/ (* x y) 2.0) (/ z 8.0)))