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

Percentage Accurate: 100.0% → 100.0%

Time: 7.9s

Alternatives: 6

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(y * N[(z * -0.5), $MachinePrecision] + N[(0.125 * x + 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. +-commutative 100.0%

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

   2. associate-+r- 100.0%

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

   3. associate-/l* 100.0%

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

   4. cancel-sign-sub-inv 100.0%

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

   5. associate-/l* 100.0%

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

   6. distribute-lft-neg-out 100.0%

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

   7. distribute-rgt-neg-out 100.0%

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

   8. +-commutative 100.0%

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

   9. associate-/l* 100.0%

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

   10. fma-define 100.0%

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

   11. neg-mul-1 100.0%

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

   12. *-commutative 100.0%

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

   13. associate-/l* 100.0%

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

   14. metadata-eval 100.0%

       \[\leadsto \mathsf{fma}\left(y, z \cdot \color{blue}{-0.5}, t + \frac{1}{8} \cdot x\right) \]

   15. +-commutative 100.0%

       \[\leadsto \mathsf{fma}\left(y, z \cdot -0.5, \color{blue}{\frac{1}{8} \cdot x + t}\right) \]

   16. fma-define 100.0%

       \[\leadsto \mathsf{fma}\left(y, z \cdot -0.5, \color{blue}{\mathsf{fma}\left(\frac{1}{8}, x, t\right)}\right) \]

   17. metadata-eval 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 87.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot z \leq -2 \cdot 10^{+18} \lor \neg \left(y \cdot z \leq 5 \cdot 10^{-34}\right):\\ \;\;\;\;t - \left(y \cdot z\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* y z) -2e+18) (not (<= (* y z) 5e-34)))
   (- 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+18) || !((y * z) <= 5e-34)) {
		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+18)) .or. (.not. ((y * z) <= 5d-34))) 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+18) || !((y * z) <= 5e-34)) {
		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+18) or not ((y * z) <= 5e-34):
		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+18) || !(Float64(y * z) <= 5e-34))
		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+18) || ~(((y * z) <= 5e-34)))
		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+18], N[Not[LessEqual[N[(y * z), $MachinePrecision], 5e-34]], $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) < -2e18 or 5.0000000000000003e-34 < (*.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. *-commutative 100.0%

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

      5. metadata-eval 100.0%

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

      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 84.6%

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

   if -2e18 < (*.f64 y z) < 5.0000000000000003e-34

   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. *-commutative 100.0%

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

      5. metadata-eval 100.0%

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

      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 95.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -2 \cdot 10^{+18} \lor \neg \left(y \cdot z \leq 5 \cdot 10^{-34}\right):\\ \;\;\;\;t - \left(y \cdot z\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -62000 \lor \neg \left(z \leq 6 \cdot 10^{+183}\right):\\ \;\;\;\;z \cdot \left(y \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -62000.0) (not (<= z 6e+183)))
   (* z (* y -0.5))
   (+ t (* 0.125 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -62000.0) || !(z <= 6e+183)) {
		tmp = z * (y * -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 ((z <= (-62000.0d0)) .or. (.not. (z <= 6d+183))) then
        tmp = z * (y * (-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 ((z <= -62000.0) || !(z <= 6e+183)) {
		tmp = z * (y * -0.5);
	} else {
		tmp = t + (0.125 * x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -62000.0) or not (z <= 6e+183):
		tmp = z * (y * -0.5)
	else:
		tmp = t + (0.125 * x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -62000.0) || !(z <= 6e+183))
		tmp = Float64(z * Float64(y * -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 ((z <= -62000.0) || ~((z <= 6e+183)))
		tmp = z * (y * -0.5);
	else
		tmp = t + (0.125 * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -62000.0], N[Not[LessEqual[z, 6e+183]], $MachinePrecision]], N[(z * N[(y * -0.5), $MachinePrecision]), $MachinePrecision], N[(t + N[(0.125 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -62000 or 5.99999999999999992e183 < 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. *-commutative 100.0%

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

      5. metadata-eval 100.0%

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

      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 81.8%

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

      1. associate-*r* 81.8%

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

      2. *-commutative 81.8%

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

      3. metadata-eval 81.8%

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

      4. div-inv 81.8%

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

      5. associate-/r/ 81.7%

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

   7. Applied egg-rr 81.7%

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

   8. Taylor expanded in t around 0 63.3%

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

      1. associate-*r* 63.3%

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

   10. Simplified 63.3%

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

   if -62000 < z < 5.99999999999999992e183

   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. *-commutative 100.0%

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

      5. metadata-eval 100.0%

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

      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 75.8%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -62000 \lor \neg \left(z \leq 6 \cdot 10^{+183}\right):\\ \;\;\;\;z \cdot \left(y \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+158}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+94}:\\ \;\;\;\;z \cdot \left(y \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -5e+158) t (if (<= t 2.3e+94) (* z (* y -0.5)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -5e+158) {
		tmp = t;
	} else if (t <= 2.3e+94) {
		tmp = z * (y * -0.5);
	} 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 <= (-5d+158)) then
        tmp = t
    else if (t <= 2.3d+94) then
        tmp = z * (y * (-0.5d0))
    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 <= -5e+158) {
		tmp = t;
	} else if (t <= 2.3e+94) {
		tmp = z * (y * -0.5);
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -5e+158:
		tmp = t
	elif t <= 2.3e+94:
		tmp = z * (y * -0.5)
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -5e+158)
		tmp = t;
	elseif (t <= 2.3e+94)
		tmp = Float64(z * Float64(y * -0.5));
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -5e+158)
		tmp = t;
	elseif (t <= 2.3e+94)
		tmp = z * (y * -0.5);
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -5e+158], t, If[LessEqual[t, 2.3e+94], N[(z * N[(y * -0.5), $MachinePrecision]), $MachinePrecision], t]]

Derivation

1. Split input into 2 regimes

2. if t < -4.9999999999999996e158 or 2.3e94 < 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. +-commutative 100.0%

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

      2. associate-+r- 100.0%

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

      3. associate-/l* 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. associate-/l* 100.0%

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

      6. distribute-lft-neg-out 100.0%

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

      7. distribute-rgt-neg-out 100.0%

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

      8. +-commutative 100.0%

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

      9. associate-/l* 100.0%

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

      10. fma-define 100.0%

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

      11. neg-mul-1 100.0%

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

      12. *-commutative 100.0%

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

      13. associate-/l* 100.0%

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

      14. metadata-eval 100.0%

          \[\leadsto \mathsf{fma}\left(y, z \cdot \color{blue}{-0.5}, t + \frac{1}{8} \cdot x\right) \]

      15. +-commutative 100.0%

          \[\leadsto \mathsf{fma}\left(y, z \cdot -0.5, \color{blue}{\frac{1}{8} \cdot x + t}\right) \]

      16. fma-define 100.0%

          \[\leadsto \mathsf{fma}\left(y, z \cdot -0.5, \color{blue}{\mathsf{fma}\left(\frac{1}{8}, x, t\right)}\right) \]

      17. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 67.3%

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

   if -4.9999999999999996e158 < t < 2.3e94

   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. *-commutative 100.0%

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

      5. metadata-eval 100.0%

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

      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 58.7%

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

      1. associate-*r* 58.7%

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

      2. *-commutative 58.7%

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

      3. metadata-eval 58.7%

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

      4. div-inv 58.7%

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

      5. associate-/r/ 58.6%

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

   7. Applied egg-rr 58.6%

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

   8. Taylor expanded in t around 0 45.9%

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

      1. associate-*r* 45.9%

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

   10. Simplified 45.9%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+158}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+94}:\\ \;\;\;\;z \cdot \left(y \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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. *-commutative 100.0%

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

   5. metadata-eval 100.0%

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

   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 6: 33.1% 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. +-commutative 100.0%

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

   2. associate-+r- 100.0%

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

   3. associate-/l* 100.0%

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

   4. cancel-sign-sub-inv 100.0%

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

   5. associate-/l* 100.0%

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

   6. distribute-lft-neg-out 100.0%

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

   7. distribute-rgt-neg-out 100.0%

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

   8. +-commutative 100.0%

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

   9. associate-/l* 100.0%

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

   10. fma-define 100.0%

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

   11. neg-mul-1 100.0%

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

   12. *-commutative 100.0%

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

   13. associate-/l* 100.0%

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

   14. metadata-eval 100.0%

       \[\leadsto \mathsf{fma}\left(y, z \cdot \color{blue}{-0.5}, t + \frac{1}{8} \cdot x\right) \]

   15. +-commutative 100.0%

       \[\leadsto \mathsf{fma}\left(y, z \cdot -0.5, \color{blue}{\frac{1}{8} \cdot x + t}\right) \]

   16. fma-define 100.0%

       \[\leadsto \mathsf{fma}\left(y, z \cdot -0.5, \color{blue}{\mathsf{fma}\left(\frac{1}{8}, x, t\right)}\right) \]

   17. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in t around inf 31.7%

   \[\leadsto \color{blue}{t} \]
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 2024165 
(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))