Main:bigenough2 from A

Percentage Accurate: 100.0% → 100.0%

Time: 5.0s

Alternatives: 8

Speedup: 1.0×

• 20.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x y z) :precision binary64 (+ x (* y (+ z x))))

C

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

Fortran

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

Java

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

Python

def code(x, y, z):
	return x + (y * (z + x))

Julia

function code(x, y, z)
	return Float64(x + Float64(y * Float64(z + x)))
end

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (+ x (* y (+ z x))))

C

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

Fortran

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

Java

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

Python

def code(x, y, z):
	return x + (y * (z + x))

Julia

function code(x, y, z)
	return Float64(x + Float64(y * Float64(z + x)))
end

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (fma y (+ x z) x))

C

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

Julia

function code(x, y, z)
	return fma(y, Float64(x + z), x)
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x + y \cdot \left(z + x\right) \]
2. Step-by-step derivation

   1. +-commutative 100.0%

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

   2. fma-define 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, z + x, x\right)} \]

   3. +-commutative 100.0%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{x + z}, x\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, x + z, x\right)} \]
4. Add Preprocessing

5. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(y, x + z, x\right) \]
6. Add Preprocessing

Alternative 2: 60.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+227}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{+112}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq -6 \cdot 10^{+38}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{-14}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-184}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-156}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 10^{-50}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+109} \lor \neg \left(y \leq 3.5 \cdot 10^{+167}\right) \land y \leq 1.2 \cdot 10^{+214}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.4e+227)
   (* y x)
   (if (<= y -1.25e+112)
     (* y z)
     (if (<= y -6e+38)
       (* y x)
       (if (<= y -2.7e-14)
         (* y z)
         (if (<= y 3e-184)
           x
           (if (<= y 9e-156)
             (* y z)
             (if (<= y 1e-50)
               x
               (if (or (<= y 9e+109)
                       (and (not (<= y 3.5e+167)) (<= y 1.2e+214)))
                 (* y z)
                 (* y x))))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.4e+227) {
		tmp = y * x;
	} else if (y <= -1.25e+112) {
		tmp = y * z;
	} else if (y <= -6e+38) {
		tmp = y * x;
	} else if (y <= -2.7e-14) {
		tmp = y * z;
	} else if (y <= 3e-184) {
		tmp = x;
	} else if (y <= 9e-156) {
		tmp = y * z;
	} else if (y <= 1e-50) {
		tmp = x;
	} else if ((y <= 9e+109) || (!(y <= 3.5e+167) && (y <= 1.2e+214))) {
		tmp = y * z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-3.4d+227)) then
        tmp = y * x
    else if (y <= (-1.25d+112)) then
        tmp = y * z
    else if (y <= (-6d+38)) then
        tmp = y * x
    else if (y <= (-2.7d-14)) then
        tmp = y * z
    else if (y <= 3d-184) then
        tmp = x
    else if (y <= 9d-156) then
        tmp = y * z
    else if (y <= 1d-50) then
        tmp = x
    else if ((y <= 9d+109) .or. (.not. (y <= 3.5d+167)) .and. (y <= 1.2d+214)) then
        tmp = y * z
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.4e+227) {
		tmp = y * x;
	} else if (y <= -1.25e+112) {
		tmp = y * z;
	} else if (y <= -6e+38) {
		tmp = y * x;
	} else if (y <= -2.7e-14) {
		tmp = y * z;
	} else if (y <= 3e-184) {
		tmp = x;
	} else if (y <= 9e-156) {
		tmp = y * z;
	} else if (y <= 1e-50) {
		tmp = x;
	} else if ((y <= 9e+109) || (!(y <= 3.5e+167) && (y <= 1.2e+214))) {
		tmp = y * z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.4e+227:
		tmp = y * x
	elif y <= -1.25e+112:
		tmp = y * z
	elif y <= -6e+38:
		tmp = y * x
	elif y <= -2.7e-14:
		tmp = y * z
	elif y <= 3e-184:
		tmp = x
	elif y <= 9e-156:
		tmp = y * z
	elif y <= 1e-50:
		tmp = x
	elif (y <= 9e+109) or (not (y <= 3.5e+167) and (y <= 1.2e+214)):
		tmp = y * z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.4e+227)
		tmp = Float64(y * x);
	elseif (y <= -1.25e+112)
		tmp = Float64(y * z);
	elseif (y <= -6e+38)
		tmp = Float64(y * x);
	elseif (y <= -2.7e-14)
		tmp = Float64(y * z);
	elseif (y <= 3e-184)
		tmp = x;
	elseif (y <= 9e-156)
		tmp = Float64(y * z);
	elseif (y <= 1e-50)
		tmp = x;
	elseif ((y <= 9e+109) || (!(y <= 3.5e+167) && (y <= 1.2e+214)))
		tmp = Float64(y * z);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.4e+227)
		tmp = y * x;
	elseif (y <= -1.25e+112)
		tmp = y * z;
	elseif (y <= -6e+38)
		tmp = y * x;
	elseif (y <= -2.7e-14)
		tmp = y * z;
	elseif (y <= 3e-184)
		tmp = x;
	elseif (y <= 9e-156)
		tmp = y * z;
	elseif (y <= 1e-50)
		tmp = x;
	elseif ((y <= 9e+109) || (~((y <= 3.5e+167)) && (y <= 1.2e+214)))
		tmp = y * z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.4e+227], N[(y * x), $MachinePrecision], If[LessEqual[y, -1.25e+112], N[(y * z), $MachinePrecision], If[LessEqual[y, -6e+38], N[(y * x), $MachinePrecision], If[LessEqual[y, -2.7e-14], N[(y * z), $MachinePrecision], If[LessEqual[y, 3e-184], x, If[LessEqual[y, 9e-156], N[(y * z), $MachinePrecision], If[LessEqual[y, 1e-50], x, If[Or[LessEqual[y, 9e+109], And[N[Not[LessEqual[y, 3.5e+167]], $MachinePrecision], LessEqual[y, 1.2e+214]]], N[(y * z), $MachinePrecision], N[(y * x), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.39999999999999989e227 or -1.25e112 < y < -6.0000000000000002e38 or 8.9999999999999992e109 < y < 3.49999999999999987e167 or 1.2e214 < y

   1. Initial program 100.0%

      \[x + y \cdot \left(z + x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 75.5%

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

      1. +-commutative 75.5%

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

   5. Simplified 75.5%

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

   6. Taylor expanded in y around inf 75.5%

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

   if -3.39999999999999989e227 < y < -1.25e112 or -6.0000000000000002e38 < y < -2.6999999999999999e-14 or 2.99999999999999991e-184 < y < 8.99999999999999971e-156 or 1.00000000000000001e-50 < y < 8.9999999999999992e109 or 3.49999999999999987e167 < y < 1.2e214

   1. Initial program 99.9%

      \[x + y \cdot \left(z + x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 100.0%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. +-commutative 100.0%

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

      7. mul-1-neg 100.0%

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

      8. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf 67.4%

      \[\leadsto \color{blue}{y \cdot z} \]

   if -2.6999999999999999e-14 < y < 2.99999999999999991e-184 or 8.99999999999999971e-156 < y < 1.00000000000000001e-50

   1. Initial program 100.0%

      \[x + y \cdot \left(z + x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 80.2%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+227}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{+112}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq -6 \cdot 10^{+38}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{-14}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-184}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-156}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 10^{-50}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+109} \lor \neg \left(y \leq 3.5 \cdot 10^{+167}\right) \land y \leq 1.2 \cdot 10^{+214}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+71} \lor \neg \left(z \leq 1.7 \cdot 10^{+35} \lor \neg \left(z \leq 3.2 \cdot 10^{+145}\right) \land z \leq 3.2 \cdot 10^{+185}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.55e+71)
         (not (or (<= z 1.7e+35) (and (not (<= z 3.2e+145)) (<= z 3.2e+185)))))
   (* y z)
   (* x (+ y 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.55e+71) || !((z <= 1.7e+35) || (!(z <= 3.2e+145) && (z <= 3.2e+185)))) {
		tmp = y * z;
	} else {
		tmp = x * (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.55d+71)) .or. (.not. (z <= 1.7d+35) .or. (.not. (z <= 3.2d+145)) .and. (z <= 3.2d+185))) then
        tmp = y * z
    else
        tmp = x * (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.55e+71) || !((z <= 1.7e+35) || (!(z <= 3.2e+145) && (z <= 3.2e+185)))) {
		tmp = y * z;
	} else {
		tmp = x * (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.55e+71) or not ((z <= 1.7e+35) or (not (z <= 3.2e+145) and (z <= 3.2e+185))):
		tmp = y * z
	else:
		tmp = x * (y + 1.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.55e+71) || !((z <= 1.7e+35) || (!(z <= 3.2e+145) && (z <= 3.2e+185))))
		tmp = Float64(y * z);
	else
		tmp = Float64(x * Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.55e+71) || ~(((z <= 1.7e+35) || (~((z <= 3.2e+145)) && (z <= 3.2e+185)))))
		tmp = y * z;
	else
		tmp = x * (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.55e+71], N[Not[Or[LessEqual[z, 1.7e+35], And[N[Not[LessEqual[z, 3.2e+145]], $MachinePrecision], LessEqual[z, 3.2e+185]]]], $MachinePrecision]], N[(y * z), $MachinePrecision], N[(x * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.55000000000000009e71 or 1.7000000000000001e35 < z < 3.20000000000000008e145 or 3.20000000000000006e185 < z

   1. Initial program 100.0%

      \[x + y \cdot \left(z + x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 100.0%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. +-commutative 100.0%

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

      7. mul-1-neg 100.0%

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

      8. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf 78.8%

      \[\leadsto \color{blue}{y \cdot z} \]

   if -1.55000000000000009e71 < z < 1.7000000000000001e35 or 3.20000000000000008e145 < z < 3.20000000000000006e185

   1. Initial program 99.9%

      \[x + y \cdot \left(z + x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 85.8%

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

      1. +-commutative 85.8%

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

   5. Simplified 85.8%

      \[\leadsto \color{blue}{x \cdot \left(y + 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+71} \lor \neg \left(z \leq 1.7 \cdot 10^{+35} \lor \neg \left(z \leq 3.2 \cdot 10^{+145}\right) \land z \leq 3.2 \cdot 10^{+185}\right):\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-14} \lor \neg \left(y \leq 3 \cdot 10^{-184} \lor \neg \left(y \leq 7.8 \cdot 10^{-140}\right) \land y \leq 9 \cdot 10^{-57}\right):\\ \;\;\;\;y \cdot \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.2e-14)
         (not (or (<= y 3e-184) (and (not (<= y 7.8e-140)) (<= y 9e-57)))))
   (* y (+ x z))
   x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.2e-14) || !((y <= 3e-184) || (!(y <= 7.8e-140) && (y <= 9e-57)))) {
		tmp = y * (x + z);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-2.2d-14)) .or. (.not. (y <= 3d-184) .or. (.not. (y <= 7.8d-140)) .and. (y <= 9d-57))) then
        tmp = y * (x + z)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.2e-14) || !((y <= 3e-184) || (!(y <= 7.8e-140) && (y <= 9e-57)))) {
		tmp = y * (x + z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.2e-14) or not ((y <= 3e-184) or (not (y <= 7.8e-140) and (y <= 9e-57))):
		tmp = y * (x + z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.2e-14) || !((y <= 3e-184) || (!(y <= 7.8e-140) && (y <= 9e-57))))
		tmp = Float64(y * Float64(x + z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.2e-14) || ~(((y <= 3e-184) || (~((y <= 7.8e-140)) && (y <= 9e-57)))))
		tmp = y * (x + z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.2e-14], N[Not[Or[LessEqual[y, 3e-184], And[N[Not[LessEqual[y, 7.8e-140]], $MachinePrecision], LessEqual[y, 9e-57]]]], $MachinePrecision]], N[(y * N[(x + z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -2.2000000000000001e-14 or 2.99999999999999991e-184 < y < 7.80000000000000038e-140 or 8.99999999999999945e-57 < y

   1. Initial program 99.9%

      \[x + y \cdot \left(z + x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 98.0%

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

      1. +-commutative 98.0%

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

      2. mul-1-neg 98.0%

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

      3. unsub-neg 98.0%

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

      4. sub-neg 98.0%

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

      5. metadata-eval 98.0%

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

      6. +-commutative 98.0%

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

      7. mul-1-neg 98.0%

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

      8. unsub-neg 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in y around inf 95.2%

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

      1. cancel-sign-sub-inv 95.2%

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

      2. metadata-eval 95.2%

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

      3. *-lft-identity 95.2%

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

      4. +-commutative 95.2%

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

   8. Simplified 95.2%

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

   if -2.2000000000000001e-14 < y < 2.99999999999999991e-184 or 7.80000000000000038e-140 < y < 8.99999999999999945e-57

   1. Initial program 100.0%

      \[x + y \cdot \left(z + x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 80.8%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-14} \lor \neg \left(y \leq 3 \cdot 10^{-184} \lor \neg \left(y \leq 7.8 \cdot 10^{-140}\right) \land y \leq 9 \cdot 10^{-57}\right):\\ \;\;\;\;y \cdot \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -102000 \lor \neg \left(y \leq 0.00132\right):\\ \;\;\;\;y \cdot \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -102000.0) (not (<= y 0.00132))) (* y (+ x z)) (+ x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -102000.0) || !(y <= 0.00132)) {
		tmp = y * (x + z);
	} else {
		tmp = x + (y * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-102000.0d0)) .or. (.not. (y <= 0.00132d0))) then
        tmp = y * (x + z)
    else
        tmp = x + (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -102000.0) || !(y <= 0.00132)) {
		tmp = y * (x + z);
	} else {
		tmp = x + (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -102000.0) or not (y <= 0.00132):
		tmp = y * (x + z)
	else:
		tmp = x + (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -102000.0) || !(y <= 0.00132))
		tmp = Float64(y * Float64(x + z));
	else
		tmp = Float64(x + Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -102000.0) || ~((y <= 0.00132)))
		tmp = y * (x + z);
	else
		tmp = x + (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -102000.0], N[Not[LessEqual[y, 0.00132]], $MachinePrecision]], N[(y * N[(x + z), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -102000 or 0.00132 < y

   1. Initial program 99.9%

      \[x + y \cdot \left(z + x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf 97.6%

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

      1. +-commutative 97.6%

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

      2. mul-1-neg 97.6%

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

      3. unsub-neg 97.6%

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

      4. sub-neg 97.6%

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

      5. metadata-eval 97.6%

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

      6. +-commutative 97.6%

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

      7. mul-1-neg 97.6%

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

      8. unsub-neg 97.6%

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

   5. Simplified 97.6%

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

   6. Taylor expanded in y around inf 99.3%

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

      1. cancel-sign-sub-inv 99.3%

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

      2. metadata-eval 99.3%

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

      3. *-lft-identity 99.3%

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

      4. +-commutative 99.3%

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

   8. Simplified 99.3%

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

   if -102000 < y < 0.00132

   1. Initial program 100.0%

      \[x + y \cdot \left(z + x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 99.3%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -102000 \lor \neg \left(y \leq 0.00132\right):\\ \;\;\;\;y \cdot \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.00132\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 0.00132))) (* y x) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.00132)) {
		tmp = y * x;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-1.0d0)) .or. (.not. (y <= 0.00132d0))) then
        tmp = y * x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.0) || !(y <= 0.00132)) {
		tmp = y * x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.0) or not (y <= 0.00132):
		tmp = y * x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 0.00132))
		tmp = Float64(y * x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 0.00132)))
		tmp = y * x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 0.00132]], $MachinePrecision]], N[(y * x), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 0.00132 < y

   1. Initial program 99.9%

      \[x + y \cdot \left(z + x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 52.2%

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

      1. +-commutative 52.2%

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

   5. Simplified 52.2%

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

   6. Taylor expanded in y around inf 51.7%

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

   if -1 < y < 0.00132

   1. Initial program 100.0%

      \[x + y \cdot \left(z + x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 71.5%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.00132\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (+ x (* y (+ x z))))

C

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

Fortran

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

Java

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

Python

def code(x, y, z):
	return x + (y * (x + z))

Julia

function code(x, y, z)
	return Float64(x + Float64(y * Float64(x + z)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x + y \cdot \left(z + x\right) \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto x + y \cdot \left(x + z\right) \]
4. Add Preprocessing

Alternative 8: 36.7% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 100.0%

   \[x + y \cdot \left(z + x\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0 37.2%

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

4. Final simplification 37.2%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))