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

Percentage Accurate: 100.0% → 100.0%

Time: 7.1s

Alternatives: 8

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: 86.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot z \leq -5 \cdot 10^{+94} \lor \neg \left(y \cdot z \leq 5 \cdot 10^{-55}\right):\\ \;\;\;\;t - \left(y \cdot z\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* y z) -5e+94) (not (<= (* y z) 5e-55)))
   (- 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) <= -5e+94) || !((y * z) <= 5e-55)) {
		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) <= (-5d+94)) .or. (.not. ((y * z) <= 5d-55))) 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) <= -5e+94) || !((y * z) <= 5e-55)) {
		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) <= -5e+94) or not ((y * z) <= 5e-55):
		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) <= -5e+94) || !(Float64(y * z) <= 5e-55))
		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) <= -5e+94) || ~(((y * z) <= 5e-55)))
		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], -5e+94], N[Not[LessEqual[N[(y * z), $MachinePrecision], 5e-55]], $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) < -5.0000000000000001e94 or 5.0000000000000002e-55 < (*.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 90.3%

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

   if -5.0000000000000001e94 < (*.f64 y z) < 5.0000000000000002e-55

   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 91.4%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -5 \cdot 10^{+94} \lor \neg \left(y \cdot z \leq 5 \cdot 10^{-55}\right):\\ \;\;\;\;t - \left(y \cdot z\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot z \leq -1:\\ \;\;\;\;0.125 \cdot x + -0.5 \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;y \cdot z \leq 5 \cdot 10^{-55}:\\ \;\;\;\;t + 0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t - \left(y \cdot z\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* y z) -1.0)
   (+ (* 0.125 x) (* -0.5 (* y z)))
   (if (<= (* y z) 5e-55) (+ t (* 0.125 x)) (- t (* (* y z) 0.5)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y * z) <= -1.0) {
		tmp = (0.125 * x) + (-0.5 * (y * z));
	} else if ((y * z) <= 5e-55) {
		tmp = t + (0.125 * x);
	} else {
		tmp = t - ((y * z) * 0.5);
	}
	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) <= (-1.0d0)) then
        tmp = (0.125d0 * x) + ((-0.5d0) * (y * z))
    else if ((y * z) <= 5d-55) then
        tmp = t + (0.125d0 * x)
    else
        tmp = t - ((y * z) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y * z) <= -1.0) {
		tmp = (0.125 * x) + (-0.5 * (y * z));
	} else if ((y * z) <= 5e-55) {
		tmp = t + (0.125 * x);
	} else {
		tmp = t - ((y * z) * 0.5);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y * z) <= -1.0:
		tmp = (0.125 * x) + (-0.5 * (y * z))
	elif (y * z) <= 5e-55:
		tmp = t + (0.125 * x)
	else:
		tmp = t - ((y * z) * 0.5)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(y * z) <= -1.0)
		tmp = Float64(Float64(0.125 * x) + Float64(-0.5 * Float64(y * z)));
	elseif (Float64(y * z) <= 5e-55)
		tmp = Float64(t + Float64(0.125 * x));
	else
		tmp = Float64(t - Float64(Float64(y * z) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y * z) <= -1.0)
		tmp = (0.125 * x) + (-0.5 * (y * z));
	elseif ((y * z) <= 5e-55)
		tmp = t + (0.125 * x);
	else
		tmp = t - ((y * z) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(y * z), $MachinePrecision], -1.0], N[(N[(0.125 * x), $MachinePrecision] + N[(-0.5 * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y * z), $MachinePrecision], 5e-55], N[(t + N[(0.125 * x), $MachinePrecision]), $MachinePrecision], N[(t - N[(N[(y * z), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y z) < -1

   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 z around inf 95.8%

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

   6. Taylor expanded in t around 0 84.1%

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

   7. Taylor expanded in z around 0 88.2%

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

   if -1 < (*.f64 y z) < 5.0000000000000002e-55

   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 96.0%

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

   if 5.0000000000000002e-55 < (*.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 89.4%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -1:\\ \;\;\;\;0.125 \cdot x + -0.5 \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;y \cdot z \leq 5 \cdot 10^{-55}:\\ \;\;\;\;t + 0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t - \left(y \cdot z\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 47.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(y \cdot -0.5\right)\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{-116}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-216}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+95}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (* y -0.5))))
   (if (<= z -2.4e-116)
     t_1
     (if (<= z 1.7e-216) t (if (<= z 1.3e+95) (* 0.125 x) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * (y * -0.5);
	double tmp;
	if (z <= -2.4e-116) {
		tmp = t_1;
	} else if (z <= 1.7e-216) {
		tmp = t;
	} else if (z <= 1.3e+95) {
		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 = z * (y * (-0.5d0))
    if (z <= (-2.4d-116)) then
        tmp = t_1
    else if (z <= 1.7d-216) then
        tmp = t
    else if (z <= 1.3d+95) 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 = z * (y * -0.5);
	double tmp;
	if (z <= -2.4e-116) {
		tmp = t_1;
	} else if (z <= 1.7e-216) {
		tmp = t;
	} else if (z <= 1.3e+95) {
		tmp = 0.125 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = z * (y * -0.5)
	tmp = 0
	if z <= -2.4e-116:
		tmp = t_1
	elif z <= 1.7e-216:
		tmp = t
	elif z <= 1.3e+95:
		tmp = 0.125 * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(y * -0.5))
	tmp = 0.0
	if (z <= -2.4e-116)
		tmp = t_1;
	elseif (z <= 1.7e-216)
		tmp = t;
	elseif (z <= 1.3e+95)
		tmp = Float64(0.125 * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = z * (y * -0.5);
	tmp = 0.0;
	if (z <= -2.4e-116)
		tmp = t_1;
	elseif (z <= 1.7e-216)
		tmp = t;
	elseif (z <= 1.3e+95)
		tmp = 0.125 * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(y * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.4e-116], t$95$1, If[LessEqual[z, 1.7e-216], t, If[LessEqual[z, 1.3e+95], N[(0.125 * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.39999999999999993e-116 or 1.29999999999999995e95 < 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 77.3%

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

   6. Taylor expanded in z around inf 75.5%

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

   7. Taylor expanded in t around 0 55.8%

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

   if -2.39999999999999993e-116 < z < 1.6999999999999999e-216

   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 51.0%

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

   if 1.6999999999999999e-216 < z < 1.29999999999999995e95

   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 z around inf 83.4%

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

   6. Taylor expanded in t around 0 51.4%

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

   7. Taylor expanded in z around 0 36.9%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-116}:\\ \;\;\;\;z \cdot \left(y \cdot -0.5\right)\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-216}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+95}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7800000 \lor \neg \left(z \leq 3.1 \cdot 10^{+123}\right):\\ \;\;\;\;z \cdot \left(y \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -7800000.0) (not (<= z 3.1e+123)))
   (* z (* y -0.5))
   (+ t (* 0.125 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -7800000.0) || !(z <= 3.1e+123)) {
		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 <= (-7800000.0d0)) .or. (.not. (z <= 3.1d+123))) 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 <= -7800000.0) || !(z <= 3.1e+123)) {
		tmp = z * (y * -0.5);
	} else {
		tmp = t + (0.125 * x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -7800000.0) or not (z <= 3.1e+123):
		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 <= -7800000.0) || !(z <= 3.1e+123))
		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 <= -7800000.0) || ~((z <= 3.1e+123)))
		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, -7800000.0], N[Not[LessEqual[z, 3.1e+123]], $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 < -7.8e6 or 3.10000000000000006e123 < 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 81.5%

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

   6. Taylor expanded in z around inf 81.5%

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

   7. Taylor expanded in t around 0 66.0%

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

   if -7.8e6 < z < 3.10000000000000006e123

   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 78.3%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7800000 \lor \neg \left(z \leq 3.1 \cdot 10^{+123}\right):\\ \;\;\;\;z \cdot \left(y \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 47.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+82} \lor \neg \left(x \leq 9.5 \cdot 10^{+160}\right):\\ \;\;\;\;0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -3.4e+82) (not (<= x 9.5e+160))) (* 0.125 x) t))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.4e+82) || !(x <= 9.5e+160)) {
		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 ((x <= (-3.4d+82)) .or. (.not. (x <= 9.5d+160))) 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 ((x <= -3.4e+82) || !(x <= 9.5e+160)) {
		tmp = 0.125 * x;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -3.4e+82) or not (x <= 9.5e+160):
		tmp = 0.125 * x
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -3.4e+82) || !(x <= 9.5e+160))
		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 ((x <= -3.4e+82) || ~((x <= 9.5e+160)))
		tmp = 0.125 * x;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -3.4e+82], N[Not[LessEqual[x, 9.5e+160]], $MachinePrecision]], N[(0.125 * x), $MachinePrecision], t]

Derivation

1. Split input into 2 regimes

2. if x < -3.39999999999999994e82 or 9.5000000000000006e160 < x

   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 z around inf 64.0%

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

   6. Taylor expanded in t around 0 57.1%

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

   7. Taylor expanded in z around 0 65.4%

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

   if -3.39999999999999994e82 < x < 9.5000000000000006e160

   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 41.2%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+82} \lor \neg \left(x \leq 9.5 \cdot 10^{+160}\right):\\ \;\;\;\;0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 8: 32.7% 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 32.0%

   \[\leadsto \color{blue}{t} \]
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 2024172 
(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))