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

Percentage Accurate: 100.0% → 100.0%

Time: 6.1s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. fma-def 100.0%

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

   3. remove-double-neg 100.0%

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

   4. fma-neg 100.0%

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

   5. metadata-eval 100.0%

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

   6. remove-double-neg 100.0%

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

   7. associate-/l* 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 100.0%

   \[\leadsto \color{blue}{\left(-0.5 \cdot \left(y \cdot z\right) + 0.125 \cdot x\right)} + t \]
6. Step-by-step derivation

   1. *-commutative 100.0%

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

   2. associate-*l* 100.0%

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

   3. metadata-eval 100.0%

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

   4. distribute-rgt-neg-in 100.0%

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

   5. fma-udef 100.0%

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

   6. distribute-rgt-neg-in 100.0%

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

   7. metadata-eval 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 71.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{-38} \lor \neg \left(z \leq 1.45 \cdot 10^{+60}\right) \land \left(z \leq 1.06 \cdot 10^{+82} \lor \neg \left(z \leq 5.2 \cdot 10^{+156}\right)\right):\\ \;\;\;\;y \cdot \left(z \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot x + t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.05e-38)
         (and (not (<= z 1.45e+60))
              (or (<= z 1.06e+82) (not (<= z 5.2e+156)))))
   (* y (* z -0.5))
   (+ (* 0.125 x) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.05e-38) || (!(z <= 1.45e+60) && ((z <= 1.06e+82) || !(z <= 5.2e+156)))) {
		tmp = y * (z * -0.5);
	} else {
		tmp = (0.125 * x) + 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 ((z <= (-2.05d-38)) .or. (.not. (z <= 1.45d+60)) .and. (z <= 1.06d+82) .or. (.not. (z <= 5.2d+156))) then
        tmp = y * (z * (-0.5d0))
    else
        tmp = (0.125d0 * x) + t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.05e-38) || (!(z <= 1.45e+60) && ((z <= 1.06e+82) || !(z <= 5.2e+156)))) {
		tmp = y * (z * -0.5);
	} else {
		tmp = (0.125 * x) + t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2.05e-38) or (not (z <= 1.45e+60) and ((z <= 1.06e+82) or not (z <= 5.2e+156))):
		tmp = y * (z * -0.5)
	else:
		tmp = (0.125 * x) + t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.05e-38) || (!(z <= 1.45e+60) && ((z <= 1.06e+82) || !(z <= 5.2e+156))))
		tmp = Float64(y * Float64(z * -0.5));
	else
		tmp = Float64(Float64(0.125 * x) + t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -2.05e-38) || (~((z <= 1.45e+60)) && ((z <= 1.06e+82) || ~((z <= 5.2e+156)))))
		tmp = y * (z * -0.5);
	else
		tmp = (0.125 * x) + t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.05e-38], And[N[Not[LessEqual[z, 1.45e+60]], $MachinePrecision], Or[LessEqual[z, 1.06e+82], N[Not[LessEqual[z, 5.2e+156]], $MachinePrecision]]]], N[(y * N[(z * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[(0.125 * x), $MachinePrecision] + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.0499999999999999e-38 or 1.45e60 < z < 1.06000000000000006e82 or 5.20000000000000037e156 < 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. sub-neg 100.0%

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

      2. fma-def 100.0%

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

      3. remove-double-neg 100.0%

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

      4. fma-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. remove-double-neg 100.0%

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

      7. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 78.2%

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

      1. *-commutative 78.2%

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

      2. associate-*l* 78.2%

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

   7. Simplified 78.2%

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

   8. Taylor expanded in y around inf 58.4%

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

      1. *-commutative 58.4%

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

      2. associate-*r* 58.4%

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

      3. *-commutative 58.4%

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

   10. Simplified 58.4%

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

   if -2.0499999999999999e-38 < z < 1.45e60 or 1.06000000000000006e82 < z < 5.20000000000000037e156

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. fma-def 100.0%

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

      3. remove-double-neg 100.0%

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

      4. fma-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. remove-double-neg 100.0%

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

      7. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 81.2%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{-38} \lor \neg \left(z \leq 1.45 \cdot 10^{+60}\right) \land \left(z \leq 1.06 \cdot 10^{+82} \lor \neg \left(z \leq 5.2 \cdot 10^{+156}\right)\right):\\ \;\;\;\;y \cdot \left(z \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot x + t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-39} \lor \neg \left(z \leq 3.1 \cdot 10^{+37}\right):\\ \;\;\;\;t + y \cdot \left(z \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot x + t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -3.4e-39) (not (<= z 3.1e+37)))
   (+ t (* y (* z -0.5)))
   (+ (* 0.125 x) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.4e-39) || !(z <= 3.1e+37)) {
		tmp = t + (y * (z * -0.5));
	} else {
		tmp = (0.125 * x) + 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 ((z <= (-3.4d-39)) .or. (.not. (z <= 3.1d+37))) then
        tmp = t + (y * (z * (-0.5d0)))
    else
        tmp = (0.125d0 * x) + t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.4e-39) || !(z <= 3.1e+37)) {
		tmp = t + (y * (z * -0.5));
	} else {
		tmp = (0.125 * x) + t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -3.4e-39) or not (z <= 3.1e+37):
		tmp = t + (y * (z * -0.5))
	else:
		tmp = (0.125 * x) + t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -3.4e-39) || !(z <= 3.1e+37))
		tmp = Float64(t + Float64(y * Float64(z * -0.5)));
	else
		tmp = Float64(Float64(0.125 * x) + t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -3.4e-39) || ~((z <= 3.1e+37)))
		tmp = t + (y * (z * -0.5));
	else
		tmp = (0.125 * x) + t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -3.4e-39], N[Not[LessEqual[z, 3.1e+37]], $MachinePrecision]], N[(t + N[(y * N[(z * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.125 * x), $MachinePrecision] + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.3999999999999999e-39 or 3.1000000000000002e37 < 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. sub-neg 100.0%

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

      2. fma-def 100.0%

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

      3. remove-double-neg 100.0%

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

      4. fma-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. remove-double-neg 100.0%

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

      7. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 78.4%

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

      1. *-commutative 78.4%

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

      2. associate-*l* 78.4%

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

   7. Simplified 78.4%

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

   if -3.3999999999999999e-39 < z < 3.1000000000000002e37

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. fma-def 100.0%

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

      3. remove-double-neg 100.0%

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

      4. fma-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. remove-double-neg 100.0%

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

      7. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 82.8%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-39} \lor \neg \left(z \leq 3.1 \cdot 10^{+37}\right):\\ \;\;\;\;t + y \cdot \left(z \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot x + t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

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 * 0.5d0)))
end function

Java

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

Python

def code(x, y, z, t):
	return t + ((0.125 * x) - (y * (z * 0.5)))

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = t + ((0.125 * x) - (y * (z * 0.5)));
end

Wolfram

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

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

   2. fma-def 100.0%

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

   3. remove-double-neg 100.0%

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

   4. fma-neg 100.0%

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

   5. metadata-eval 100.0%

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

   6. remove-double-neg 100.0%

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

   7. associate-/l* 99.9%

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

3. Simplified 99.9%

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

   1. clear-num 99.9%

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

   2. associate-/r/ 99.9%

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

   3. clear-num 100.0%

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

   4. div-inv 100.0%

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

   5. metadata-eval 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

   \[\leadsto t + \left(0.125 \cdot x - y \cdot \left(z \cdot 0.5\right)\right) \]
8. Add Preprocessing

Alternative 5: 51.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{+42}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+43}:\\ \;\;\;\;y \cdot \left(z \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -4e+42) t (if (<= t 8.5e+43) (* y (* z -0.5)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -4e+42) {
		tmp = t;
	} else if (t <= 8.5e+43) {
		tmp = y * (z * -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 <= (-4d+42)) then
        tmp = t
    else if (t <= 8.5d+43) then
        tmp = y * (z * (-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 <= -4e+42) {
		tmp = t;
	} else if (t <= 8.5e+43) {
		tmp = y * (z * -0.5);
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -4e+42:
		tmp = t
	elif t <= 8.5e+43:
		tmp = y * (z * -0.5)
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -4e+42)
		tmp = t;
	elseif (t <= 8.5e+43)
		tmp = Float64(y * Float64(z * -0.5));
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -4e+42)
		tmp = t;
	elseif (t <= 8.5e+43)
		tmp = y * (z * -0.5);
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -4e+42], t, If[LessEqual[t, 8.5e+43], N[(y * N[(z * -0.5), $MachinePrecision]), $MachinePrecision], t]]

Derivation

1. Split input into 2 regimes

2. if t < -4.00000000000000018e42 or 8.5e43 < 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. sub-neg 100.0%

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

      2. fma-def 100.0%

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

      3. remove-double-neg 100.0%

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

      4. fma-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. remove-double-neg 100.0%

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

      7. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 83.7%

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

      1. *-commutative 83.7%

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

      2. associate-*l* 83.7%

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

   7. Simplified 83.7%

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

   8. Taylor expanded in y around 0 66.6%

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

   if -4.00000000000000018e42 < t < 8.5e43

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. fma-def 100.0%

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

      3. remove-double-neg 100.0%

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

      4. fma-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. remove-double-neg 100.0%

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

      7. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 58.3%

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

      1. *-commutative 58.3%

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

      2. associate-*l* 58.3%

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

   7. Simplified 58.3%

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

   8. Taylor expanded in y around inf 49.0%

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

      1. *-commutative 49.0%

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

      2. associate-*r* 49.0%

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

      3. *-commutative 49.0%

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

   10. Simplified 49.0%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{+42}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+43}:\\ \;\;\;\;y \cdot \left(z \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 32.4% 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. sub-neg 100.0%

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

   2. fma-def 100.0%

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

   3. remove-double-neg 100.0%

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

   4. fma-neg 100.0%

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

   5. metadata-eval 100.0%

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

   6. remove-double-neg 100.0%

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

   7. associate-/l* 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 68.3%

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

   1. *-commutative 68.3%

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

   2. associate-*l* 68.3%

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

7. Simplified 68.3%

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

8. Taylor expanded in y around 0 33.3%

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

9. Final simplification 33.3%

   \[\leadsto t \]
10. Add Preprocessing

Developer target: 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 2024010 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (- (+ (/ x 8.0) t) (* (/ z 2.0) y))

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