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

Percentage Accurate: 100.0% → 100.0%

Time: 7.2s

Alternatives: 7

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ \left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- (* (/ 1.0 8.0) x) (/ (* y z) 2.0)) t))

C

double code(double x, double y, double z, double t) {
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((1.0d0 / 8.0d0) * x) - ((y * z) / 2.0d0)) + t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t;
}

Python

def code(x, y, z, t):
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(1.0 / 8.0) * x) - Float64(Float64(y * z) / 2.0)) + t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 / 8.0), $MachinePrecision] * x), $MachinePrecision] - N[(N[(y * z), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- (* (/ 1.0 8.0) x) (/ (* y z) 2.0)) t))

C

double code(double x, double y, double z, double t) {
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((1.0d0 / 8.0d0) * x) - ((y * z) / 2.0d0)) + t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t;
}

Python

def code(x, y, z, t):
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(1.0 / 8.0) * x) - Float64(Float64(y * z) / 2.0)) + t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 / 8.0), $MachinePrecision] * x), $MachinePrecision] - N[(N[(y * z), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.125, x, \mathsf{fma}\left(y, z \cdot -0.5, t\right)\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (fma 0.125 x (fma y (* z -0.5) t)))

C

double code(double x, double y, double z, double t) {
	return fma(0.125, x, fma(y, (z * -0.5), t));
}

Julia

function code(x, y, z, t)
	return fma(0.125, x, fma(y, Float64(z * -0.5), t))
end

Wolfram

code[x_, y_, z_, t_] := N[(0.125 * x + N[(y * N[(z * -0.5), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
2. Step-by-step derivation

   1. associate-+l- 100.0%

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

   2. fma-neg 100.0%

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

   3. metadata-eval 100.0%

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

   4. sub-neg 100.0%

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

   5. distribute-neg-in 100.0%

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

   6. distribute-frac-neg 100.0%

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

   7. distribute-rgt-neg-out 100.0%

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

   8. remove-double-neg 100.0%

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

   9. associate-/l* 100.0%

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

   10. fma-define 100.0%

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

   11. neg-mul-1 100.0%

       \[\leadsto \mathsf{fma}\left(0.125, x, \mathsf{fma}\left(y, \frac{\color{blue}{-1 \cdot z}}{2}, t\right)\right) \]

   12. *-commutative 100.0%

       \[\leadsto \mathsf{fma}\left(0.125, x, \mathsf{fma}\left(y, \frac{\color{blue}{z \cdot -1}}{2}, t\right)\right) \]

   13. associate-/l* 100.0%

       \[\leadsto \mathsf{fma}\left(0.125, x, \mathsf{fma}\left(y, \color{blue}{z \cdot \frac{-1}{2}}, t\right)\right) \]

   14. metadata-eval 100.0%

       \[\leadsto \mathsf{fma}\left(0.125, x, \mathsf{fma}\left(y, z \cdot \color{blue}{-0.5}, t\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, x, \mathsf{fma}\left(y, z \cdot -0.5, t\right)\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 87.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot z \leq -2 \cdot 10^{-38} \lor \neg \left(y \cdot z \leq 5 \cdot 10^{+114}\right):\\ \;\;\;\;t - \left(y \cdot z\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* y z) -2e-38) (not (<= (* y z) 5e+114)))
   (- t (* (* y z) 0.5))
   (+ t (* 0.125 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((y * z) <= -2e-38) || !((y * z) <= 5e+114)) {
		tmp = t - ((y * z) * 0.5);
	} else {
		tmp = t + (0.125 * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((y * z) <= (-2d-38)) .or. (.not. ((y * z) <= 5d+114))) then
        tmp = t - ((y * z) * 0.5d0)
    else
        tmp = t + (0.125d0 * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((y * z) <= -2e-38) || !((y * z) <= 5e+114)) {
		tmp = t - ((y * z) * 0.5);
	} else {
		tmp = t + (0.125 * x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((y * z) <= -2e-38) or not ((y * z) <= 5e+114):
		tmp = t - ((y * z) * 0.5)
	else:
		tmp = t + (0.125 * x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(y * z) <= -2e-38) || !(Float64(y * z) <= 5e+114))
		tmp = Float64(t - Float64(Float64(y * z) * 0.5));
	else
		tmp = Float64(t + Float64(0.125 * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((y * z) <= -2e-38) || ~(((y * z) <= 5e+114)))
		tmp = t - ((y * z) * 0.5);
	else
		tmp = t + (0.125 * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(y * z), $MachinePrecision], -2e-38], N[Not[LessEqual[N[(y * z), $MachinePrecision], 5e+114]], $MachinePrecision]], N[(t - N[(N[(y * z), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(t + N[(0.125 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -1.9999999999999999e-38 or 5.0000000000000001e114 < (*.f64 y z)

   1. Initial program 100.0%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l- 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 94.6%

      \[\leadsto \color{blue}{t - 0.5 \cdot \left(y \cdot z\right)} \]

   if -1.9999999999999999e-38 < (*.f64 y z) < 5.0000000000000001e114

   1. Initial program 100.0%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l- 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 89.9%

      \[\leadsto \color{blue}{0.125 \cdot x} + t \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -2 \cdot 10^{-38} \lor \neg \left(y \cdot z \leq 5 \cdot 10^{+114}\right):\\ \;\;\;\;t - \left(y \cdot z\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(z \cdot -0.5\right)\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-173}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+85}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (* z -0.5))))
   (if (<= z -1.3e-11)
     t_1
     (if (<= z 1.6e-173) t (if (<= z 1.7e+85) (* 0.125 x) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (z * -0.5);
	double tmp;
	if (z <= -1.3e-11) {
		tmp = t_1;
	} else if (z <= 1.6e-173) {
		tmp = t;
	} else if (z <= 1.7e+85) {
		tmp = 0.125 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y * (z * (-0.5d0))
    if (z <= (-1.3d-11)) then
        tmp = t_1
    else if (z <= 1.6d-173) then
        tmp = t
    else if (z <= 1.7d+85) then
        tmp = 0.125d0 * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = y * (z * -0.5);
	double tmp;
	if (z <= -1.3e-11) {
		tmp = t_1;
	} else if (z <= 1.6e-173) {
		tmp = t;
	} else if (z <= 1.7e+85) {
		tmp = 0.125 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y * (z * -0.5)
	tmp = 0
	if z <= -1.3e-11:
		tmp = t_1
	elif z <= 1.6e-173:
		tmp = t
	elif z <= 1.7e+85:
		tmp = 0.125 * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(z * -0.5))
	tmp = 0.0
	if (z <= -1.3e-11)
		tmp = t_1;
	elseif (z <= 1.6e-173)
		tmp = t;
	elseif (z <= 1.7e+85)
		tmp = Float64(0.125 * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y * (z * -0.5);
	tmp = 0.0;
	if (z <= -1.3e-11)
		tmp = t_1;
	elseif (z <= 1.6e-173)
		tmp = t;
	elseif (z <= 1.7e+85)
		tmp = 0.125 * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(z * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.3e-11], t$95$1, If[LessEqual[z, 1.6e-173], t, If[LessEqual[z, 1.7e+85], N[(0.125 * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.3e-11 or 1.7000000000000002e85 < z

   1. Initial program 100.0%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l- 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 84.7%

      \[\leadsto \color{blue}{t - 0.5 \cdot \left(y \cdot z\right)} \]

   6. Taylor expanded in t around 0 61.3%

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

      1. *-commutative 61.3%

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

      2. associate-*r* 61.3%

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

      3. *-commutative 61.3%

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

   8. Simplified 61.3%

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

   if -1.3e-11 < z < 1.6e-173

   1. Initial program 100.0%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

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

      2. fma-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. distribute-rgt-neg-out 100.0%

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

      8. remove-double-neg 100.0%

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

      9. associate-/l* 100.0%

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

      10. fma-define 100.0%

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

      11. neg-mul-1 100.0%

          \[\leadsto \mathsf{fma}\left(0.125, x, \mathsf{fma}\left(y, \frac{\color{blue}{-1 \cdot z}}{2}, t\right)\right) \]

      12. *-commutative 100.0%

          \[\leadsto \mathsf{fma}\left(0.125, x, \mathsf{fma}\left(y, \frac{\color{blue}{z \cdot -1}}{2}, t\right)\right) \]

      13. associate-/l* 100.0%

          \[\leadsto \mathsf{fma}\left(0.125, x, \mathsf{fma}\left(y, \color{blue}{z \cdot \frac{-1}{2}}, t\right)\right) \]

      14. metadata-eval 100.0%

          \[\leadsto \mathsf{fma}\left(0.125, x, \mathsf{fma}\left(y, z \cdot \color{blue}{-0.5}, t\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, x, \mathsf{fma}\left(y, z \cdot -0.5, t\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 46.5%

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

   if 1.6e-173 < z < 1.7000000000000002e85

   1. Initial program 100.0%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l- 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 77.2%

      \[\leadsto \color{blue}{0.125 \cdot x} + t \]

   6. Taylor expanded in x around inf 50.7%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-11}:\\ \;\;\;\;y \cdot \left(z \cdot -0.5\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-173}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+85}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{+136} \lor \neg \left(y \leq 6.8 \cdot 10^{+15}\right):\\ \;\;\;\;y \cdot \left(z \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -4.3e+136) (not (<= y 6.8e+15)))
   (* y (* z -0.5))
   (+ t (* 0.125 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -4.3e+136) || !(y <= 6.8e+15)) {
		tmp = y * (z * -0.5);
	} else {
		tmp = t + (0.125 * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-4.3d+136)) .or. (.not. (y <= 6.8d+15))) then
        tmp = y * (z * (-0.5d0))
    else
        tmp = t + (0.125d0 * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -4.3e+136) || !(y <= 6.8e+15)) {
		tmp = y * (z * -0.5);
	} else {
		tmp = t + (0.125 * x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -4.3e+136) or not (y <= 6.8e+15):
		tmp = y * (z * -0.5)
	else:
		tmp = t + (0.125 * x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -4.3e+136) || !(y <= 6.8e+15))
		tmp = Float64(y * Float64(z * -0.5));
	else
		tmp = Float64(t + Float64(0.125 * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -4.3e+136) || ~((y <= 6.8e+15)))
		tmp = y * (z * -0.5);
	else
		tmp = t + (0.125 * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -4.3e+136], N[Not[LessEqual[y, 6.8e+15]], $MachinePrecision]], N[(y * N[(z * -0.5), $MachinePrecision]), $MachinePrecision], N[(t + N[(0.125 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.2999999999999999e136 or 6.8e15 < y

   1. Initial program 100.0%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l- 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 86.9%

      \[\leadsto \color{blue}{t - 0.5 \cdot \left(y \cdot z\right)} \]

   6. Taylor expanded in t around 0 68.0%

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

      1. *-commutative 68.0%

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

      2. associate-*r* 68.0%

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

      3. *-commutative 68.0%

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

   8. Simplified 68.0%

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

   if -4.2999999999999999e136 < y < 6.8e15

   1. Initial program 100.0%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l- 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 78.6%

      \[\leadsto \color{blue}{0.125 \cdot x} + t \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{+136} \lor \neg \left(y \leq 6.8 \cdot 10^{+15}\right):\\ \;\;\;\;y \cdot \left(z \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.06 \cdot 10^{+23}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 0.028:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.06e+23) t (if (<= t 0.028) (* 0.125 x) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.06e+23) {
		tmp = t;
	} else if (t <= 0.028) {
		tmp = 0.125 * x;
	} else {
		tmp = t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.06d+23)) then
        tmp = t
    else if (t <= 0.028d0) then
        tmp = 0.125d0 * x
    else
        tmp = t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.06e+23) {
		tmp = t;
	} else if (t <= 0.028) {
		tmp = 0.125 * x;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.06e+23:
		tmp = t
	elif t <= 0.028:
		tmp = 0.125 * x
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.06e+23)
		tmp = t;
	elseif (t <= 0.028)
		tmp = Float64(0.125 * x);
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.06e+23)
		tmp = t;
	elseif (t <= 0.028)
		tmp = 0.125 * x;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.06e+23], t, If[LessEqual[t, 0.028], N[(0.125 * x), $MachinePrecision], t]]

Derivation

1. Split input into 2 regimes

2. if t < -1.06e23 or 0.0280000000000000006 < t

   1. Initial program 100.0%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

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

      2. fma-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. distribute-frac-neg 100.0%

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

      7. distribute-rgt-neg-out 100.0%

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

      8. remove-double-neg 100.0%

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

      9. associate-/l* 100.0%

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

      10. fma-define 100.0%

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

      11. neg-mul-1 100.0%

          \[\leadsto \mathsf{fma}\left(0.125, x, \mathsf{fma}\left(y, \frac{\color{blue}{-1 \cdot z}}{2}, t\right)\right) \]

      12. *-commutative 100.0%

          \[\leadsto \mathsf{fma}\left(0.125, x, \mathsf{fma}\left(y, \frac{\color{blue}{z \cdot -1}}{2}, t\right)\right) \]

      13. associate-/l* 100.0%

          \[\leadsto \mathsf{fma}\left(0.125, x, \mathsf{fma}\left(y, \color{blue}{z \cdot \frac{-1}{2}}, t\right)\right) \]

      14. metadata-eval 100.0%

          \[\leadsto \mathsf{fma}\left(0.125, x, \mathsf{fma}\left(y, z \cdot \color{blue}{-0.5}, t\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, x, \mathsf{fma}\left(y, z \cdot -0.5, t\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 57.3%

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

   if -1.06e23 < t < 0.0280000000000000006

   1. Initial program 100.0%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l- 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 53.4%

      \[\leadsto \color{blue}{0.125 \cdot x} + t \]

   6. Taylor expanded in x around inf 48.4%

      \[\leadsto \color{blue}{0.125 \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = t + ((0.125d0 * x) - (y * (z / 2.0d0)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return t + ((0.125 * x) - (y * (z / 2.0)));
}

Python

def code(x, y, z, t):
	return t + ((0.125 * x) - (y * (z / 2.0)))

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = t + ((0.125 * x) - (y * (z / 2.0)));
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
2. Step-by-step derivation

   1. associate-+l- 100.0%

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

   2. *-commutative 100.0%

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

   3. associate-+l- 100.0%

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

   4. metadata-eval 100.0%

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

   5. *-commutative 100.0%

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

   6. associate-/l* 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 7: 32.5% accurate, 13.0× speedup

Math

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

FPCore

(FPCore (x y z t) :precision binary64 t)

C

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

Fortran

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

Java

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

Python

def code(x, y, z, t):
	return t

Julia

function code(x, y, z, t)
	return t
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := t

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
2. Step-by-step derivation

   1. associate-+l- 100.0%

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

   2. fma-neg 100.0%

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

   3. metadata-eval 100.0%

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

   4. sub-neg 100.0%

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

   5. distribute-neg-in 100.0%

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

   6. distribute-frac-neg 100.0%

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

   7. distribute-rgt-neg-out 100.0%

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

   8. remove-double-neg 100.0%

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

   9. associate-/l* 100.0%

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

   10. fma-define 100.0%

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

   11. neg-mul-1 100.0%

       \[\leadsto \mathsf{fma}\left(0.125, x, \mathsf{fma}\left(y, \frac{\color{blue}{-1 \cdot z}}{2}, t\right)\right) \]

   12. *-commutative 100.0%

       \[\leadsto \mathsf{fma}\left(0.125, x, \mathsf{fma}\left(y, \frac{\color{blue}{z \cdot -1}}{2}, t\right)\right) \]

   13. associate-/l* 100.0%

       \[\leadsto \mathsf{fma}\left(0.125, x, \mathsf{fma}\left(y, \color{blue}{z \cdot \frac{-1}{2}}, t\right)\right) \]

   14. metadata-eval 100.0%

       \[\leadsto \mathsf{fma}\left(0.125, x, \mathsf{fma}\left(y, z \cdot \color{blue}{-0.5}, t\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, x, \mathsf{fma}\left(y, z \cdot -0.5, t\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in t around inf 32.7%

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

Developer Target 1: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\frac{x}{8} + t\right) - \frac{z}{2} \cdot y \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x / 8.0d0) + t) - ((z / 2.0d0) * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024133 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (+ (/ x 8) t) (* (/ z 2) y)))

  (+ (- (* (/ 1.0 8.0) x) (/ (* y z) 2.0)) t))