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

Percentage Accurate: 100.0% → 100.0%

Time: 4.1s

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: 66.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-182}:\\ \;\;\;\;z \cdot -0.125\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+55}:\\ \;\;\;\;0.5 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot -0.125\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -7e-182)
   (* z -0.125)
   (if (<= z 2.9e+55) (* 0.5 (* x y)) (* z -0.125))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -7e-182) {
		tmp = z * -0.125;
	} else if (z <= 2.9e+55) {
		tmp = 0.5 * (x * y);
	} 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 (z <= (-7d-182)) then
        tmp = z * (-0.125d0)
    else if (z <= 2.9d+55) then
        tmp = 0.5d0 * (x * y)
    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 (z <= -7e-182) {
		tmp = z * -0.125;
	} else if (z <= 2.9e+55) {
		tmp = 0.5 * (x * y);
	} else {
		tmp = z * -0.125;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -7e-182:
		tmp = z * -0.125
	elif z <= 2.9e+55:
		tmp = 0.5 * (x * y)
	else:
		tmp = z * -0.125
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -7e-182)
		tmp = Float64(z * -0.125);
	elseif (z <= 2.9e+55)
		tmp = Float64(0.5 * Float64(x * y));
	else
		tmp = Float64(z * -0.125);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -7e-182)
		tmp = z * -0.125;
	elseif (z <= 2.9e+55)
		tmp = 0.5 * (x * y);
	else
		tmp = z * -0.125;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -7e-182], N[(z * -0.125), $MachinePrecision], If[LessEqual[z, 2.9e+55], N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision], N[(z * -0.125), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.99999999999999966e-182 or 2.8999999999999999e55 < z

   1. Initial program 100.0%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 76.3%

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

   if -6.99999999999999966e-182 < z < 2.8999999999999999e55

   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. Step-by-step derivation

      1. frac-2neg 99.8%

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

      2. clear-num 99.8%

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

      3. frac-sub 68.6%

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

      4. distribute-neg-frac 68.6%

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

      5. metadata-eval 68.6%

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

      6. distribute-neg-frac 68.6%

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

      7. metadata-eval 68.6%

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

   5. Applied egg-rr 68.6%

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

   6. Taylor expanded in x around inf 78.5%

      \[\leadsto \color{blue}{0.5 \cdot \left(y \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-182}:\\ \;\;\;\;z \cdot -0.125\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+55}:\\ \;\;\;\;0.5 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot -0.125\\ \end{array} \]

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* 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

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: 51.6% 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* 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 56.3%

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

5. Final simplification 56.3%

   \[\leadsto z \cdot -0.125 \]

Reproduce

herbie shell --seed 2023199 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, D"
  :precision binary64
  (- (/ (* x y) 2.0) (/ z 8.0)))