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

Percentage Accurate: 100.0% → 100.0%

Time: 2.5s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*l/ 100.0%

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

   3. fma-neg 100.0%

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

   4. distribute-frac-neg 100.0%

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

   5. neg-mul-1 100.0%

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

   6. associate-/l* 99.9%

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

   7. associate-/r/ 100.0%

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

   8. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 78.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot x \leq -1.65 \cdot 10^{+43} \lor \neg \left(y \cdot x \leq 3 \cdot 10^{-45}\right) \land \left(y \cdot x \leq 8600000 \lor \neg \left(y \cdot x \leq 5 \cdot 10^{+75}\right)\right):\\ \;\;\;\;\left(y \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* y x) -1.65e+43)
         (and (not (<= (* y x) 3e-45))
              (or (<= (* y x) 8600000.0) (not (<= (* y x) 5e+75)))))
   (* (* y x) 0.5)
   (* -0.125 z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((y * x) <= -1.65e+43) || (!((y * x) <= 3e-45) && (((y * x) <= 8600000.0) || !((y * x) <= 5e+75)))) {
		tmp = (y * x) * 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 (((y * x) <= (-1.65d+43)) .or. (.not. ((y * x) <= 3d-45)) .and. ((y * x) <= 8600000.0d0) .or. (.not. ((y * x) <= 5d+75))) then
        tmp = (y * x) * 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 (((y * x) <= -1.65e+43) || (!((y * x) <= 3e-45) && (((y * x) <= 8600000.0) || !((y * x) <= 5e+75)))) {
		tmp = (y * x) * 0.5;
	} else {
		tmp = -0.125 * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((y * x) <= -1.65e+43) or (not ((y * x) <= 3e-45) and (((y * x) <= 8600000.0) or not ((y * x) <= 5e+75))):
		tmp = (y * x) * 0.5
	else:
		tmp = -0.125 * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(y * x) <= -1.65e+43) || (!(Float64(y * x) <= 3e-45) && ((Float64(y * x) <= 8600000.0) || !(Float64(y * x) <= 5e+75))))
		tmp = Float64(Float64(y * x) * 0.5);
	else
		tmp = Float64(-0.125 * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((y * x) <= -1.65e+43) || (~(((y * x) <= 3e-45)) && (((y * x) <= 8600000.0) || ~(((y * x) <= 5e+75)))))
		tmp = (y * x) * 0.5;
	else
		tmp = -0.125 * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(y * x), $MachinePrecision], -1.65e+43], And[N[Not[LessEqual[N[(y * x), $MachinePrecision], 3e-45]], $MachinePrecision], Or[LessEqual[N[(y * x), $MachinePrecision], 8600000.0], N[Not[LessEqual[N[(y * x), $MachinePrecision], 5e+75]], $MachinePrecision]]]], N[(N[(y * x), $MachinePrecision] * 0.5), $MachinePrecision], N[(-0.125 * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.6500000000000001e43 or 3.00000000000000011e-45 < (*.f64 x y) < 8.6e6 or 5.0000000000000002e75 < (*.f64 x y)

   1. Initial program 99.3%

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

      1. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 83.2%

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

   if -1.6500000000000001e43 < (*.f64 x y) < 3.00000000000000011e-45 or 8.6e6 < (*.f64 x y) < 5.0000000000000002e75

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 77.6%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -1.65 \cdot 10^{+43} \lor \neg \left(y \cdot x \leq 3 \cdot 10^{-45}\right) \land \left(y \cdot x \leq 8600000 \lor \neg \left(y \cdot x \leq 5 \cdot 10^{+75}\right)\right):\\ \;\;\;\;\left(y \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot z\\ \end{array} \]

Alternative 3: 78.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot 0.5\\ \mathbf{if}\;y \cdot x \leq -2 \cdot 10^{+48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \cdot x \leq 2 \cdot 10^{-45}:\\ \;\;\;\;-0.125 \cdot z\\ \mathbf{elif}\;y \cdot x \leq 1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \cdot x \leq 5 \cdot 10^{+75}:\\ \;\;\;\;-0.125 \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* y x) 0.5)))
   (if (<= (* y x) -2e+48)
     t_0
     (if (<= (* y x) 2e-45)
       (* -0.125 z)
       (if (<= (* y x) 1.0)
         t_0
         (if (<= (* y x) 5e+75) (* -0.125 z) (* x (* y 0.5))))))))

C

double code(double x, double y, double z) {
	double t_0 = (y * x) * 0.5;
	double tmp;
	if ((y * x) <= -2e+48) {
		tmp = t_0;
	} else if ((y * x) <= 2e-45) {
		tmp = -0.125 * z;
	} else if ((y * x) <= 1.0) {
		tmp = t_0;
	} else if ((y * x) <= 5e+75) {
		tmp = -0.125 * z;
	} else {
		tmp = x * (y * 0.5);
	}
	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) :: tmp
    t_0 = (y * x) * 0.5d0
    if ((y * x) <= (-2d+48)) then
        tmp = t_0
    else if ((y * x) <= 2d-45) then
        tmp = (-0.125d0) * z
    else if ((y * x) <= 1.0d0) then
        tmp = t_0
    else if ((y * x) <= 5d+75) then
        tmp = (-0.125d0) * z
    else
        tmp = x * (y * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y * x) * 0.5;
	double tmp;
	if ((y * x) <= -2e+48) {
		tmp = t_0;
	} else if ((y * x) <= 2e-45) {
		tmp = -0.125 * z;
	} else if ((y * x) <= 1.0) {
		tmp = t_0;
	} else if ((y * x) <= 5e+75) {
		tmp = -0.125 * z;
	} else {
		tmp = x * (y * 0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y * x) * 0.5
	tmp = 0
	if (y * x) <= -2e+48:
		tmp = t_0
	elif (y * x) <= 2e-45:
		tmp = -0.125 * z
	elif (y * x) <= 1.0:
		tmp = t_0
	elif (y * x) <= 5e+75:
		tmp = -0.125 * z
	else:
		tmp = x * (y * 0.5)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * x) * 0.5)
	tmp = 0.0
	if (Float64(y * x) <= -2e+48)
		tmp = t_0;
	elseif (Float64(y * x) <= 2e-45)
		tmp = Float64(-0.125 * z);
	elseif (Float64(y * x) <= 1.0)
		tmp = t_0;
	elseif (Float64(y * x) <= 5e+75)
		tmp = Float64(-0.125 * z);
	else
		tmp = Float64(x * Float64(y * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y * x) * 0.5;
	tmp = 0.0;
	if ((y * x) <= -2e+48)
		tmp = t_0;
	elseif ((y * x) <= 2e-45)
		tmp = -0.125 * z;
	elseif ((y * x) <= 1.0)
		tmp = t_0;
	elseif ((y * x) <= 5e+75)
		tmp = -0.125 * z;
	else
		tmp = x * (y * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[N[(y * x), $MachinePrecision], -2e+48], t$95$0, If[LessEqual[N[(y * x), $MachinePrecision], 2e-45], N[(-0.125 * z), $MachinePrecision], If[LessEqual[N[(y * x), $MachinePrecision], 1.0], t$95$0, If[LessEqual[N[(y * x), $MachinePrecision], 5e+75], N[(-0.125 * z), $MachinePrecision], N[(x * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2.00000000000000009e48 or 1.99999999999999997e-45 < (*.f64 x y) < 1

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 82.5%

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

   if -2.00000000000000009e48 < (*.f64 x y) < 1.99999999999999997e-45 or 1 < (*.f64 x y) < 5.0000000000000002e75

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 77.6%

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

   if 5.0000000000000002e75 < (*.f64 x y)

   1. Initial program 98.3%

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

      1. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 84.2%

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

      1. expm1-log1p-u 79.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot \left(x \cdot y\right)\right)\right)} \]

      2. expm1-udef 79.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot \left(x \cdot y\right)\right)} - 1} \]

      3. log1p-udef 79.0%

         \[\leadsto e^{\color{blue}{\log \left(1 + 0.5 \cdot \left(x \cdot y\right)\right)}} - 1 \]

      4. add-exp-log 84.2%

         \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(x \cdot y\right)\right)} - 1 \]

   6. Applied egg-rr 84.2%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \left(x \cdot y\right)\right) - 1} \]
   7. Step-by-step derivation

      1. +-commutative 84.2%

         \[\leadsto \color{blue}{\left(0.5 \cdot \left(x \cdot y\right) + 1\right)} - 1 \]

      2. associate--l+ 84.2%

         \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot y\right) + \left(1 - 1\right)} \]

      3. metadata-eval 84.2%

         \[\leadsto 0.5 \cdot \left(x \cdot y\right) + \color{blue}{0} \]

      4. *-commutative 84.2%

         \[\leadsto 0.5 \cdot \color{blue}{\left(y \cdot x\right)} + 0 \]

      5. associate-*r* 85.8%

         \[\leadsto \color{blue}{\left(0.5 \cdot y\right) \cdot x} + 0 \]

   8. Simplified 85.8%

      \[\leadsto \color{blue}{\left(0.5 \cdot y\right) \cdot x + 0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -2 \cdot 10^{+48}:\\ \;\;\;\;\left(y \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;y \cdot x \leq 2 \cdot 10^{-45}:\\ \;\;\;\;-0.125 \cdot z\\ \mathbf{elif}\;y \cdot x \leq 1:\\ \;\;\;\;\left(y \cdot x\right) \cdot 0.5\\ \mathbf{elif}\;y \cdot x \leq 5 \cdot 10^{+75}:\\ \;\;\;\;-0.125 \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot 0.5\right)\\ \end{array} \]

Alternative 4: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (y * (x / 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 = (y * (x / 2.0d0)) - (z / 8.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate-*l/ 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

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 99.7%

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

   1. associate-*l/ 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 50.2%

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

5. Final simplification 50.2%

   \[\leadsto -0.125 \cdot z \]

Reproduce

herbie shell --seed 2023293 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, D"
  :precision binary64
  (- (/ (* x y) 2.0) (/ z 8.0)))