Main:bigenough2 from A

Percentage Accurate: 100.0% → 100.0%

Time: 4.7s

Alternatives: 5

Speedup: 1.2×

• 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, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 82.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z + x\right) \cdot y\\ \mathbf{if}\;y \leq -1.4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{elif}\;y \leq -1.85 \cdot 10^{-51}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-115}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{-81}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{-56}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+264}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ z x) y)))
   (if (<= y -1.4)
     t_0
     (if (<= y -3.6e-40)
       (fma y x x)
       (if (<= y -1.85e-51)
         (* z y)
         (if (<= y 9e-115)
           (fma y x x)
           (if (<= y 1.02e-81)
             (* z y)
             (if (<= y 7.4e-56)
               (fma y x x)
               (if (<= y 1.9e+264) t_0 (* y x))))))))))

C

double code(double x, double y, double z) {
	double t_0 = (z + x) * y;
	double tmp;
	if (y <= -1.4) {
		tmp = t_0;
	} else if (y <= -3.6e-40) {
		tmp = fma(y, x, x);
	} else if (y <= -1.85e-51) {
		tmp = z * y;
	} else if (y <= 9e-115) {
		tmp = fma(y, x, x);
	} else if (y <= 1.02e-81) {
		tmp = z * y;
	} else if (y <= 7.4e-56) {
		tmp = fma(y, x, x);
	} else if (y <= 1.9e+264) {
		tmp = t_0;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z + x) * y)
	tmp = 0.0
	if (y <= -1.4)
		tmp = t_0;
	elseif (y <= -3.6e-40)
		tmp = fma(y, x, x);
	elseif (y <= -1.85e-51)
		tmp = Float64(z * y);
	elseif (y <= 9e-115)
		tmp = fma(y, x, x);
	elseif (y <= 1.02e-81)
		tmp = Float64(z * y);
	elseif (y <= 7.4e-56)
		tmp = fma(y, x, x);
	elseif (y <= 1.9e+264)
		tmp = t_0;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z + x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -1.4], t$95$0, If[LessEqual[y, -3.6e-40], N[(y * x + x), $MachinePrecision], If[LessEqual[y, -1.85e-51], N[(z * y), $MachinePrecision], If[LessEqual[y, 9e-115], N[(y * x + x), $MachinePrecision], If[LessEqual[y, 1.02e-81], N[(z * y), $MachinePrecision], If[LessEqual[y, 7.4e-56], N[(y * x + x), $MachinePrecision], If[LessEqual[y, 1.9e+264], t$95$0, N[(y * x), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.3999999999999999 or 7.4000000000000004e-56 < y < 1.9000000000000001e264

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in y around -inf

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-out N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 97.9

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

   7. Applied rewrites 97.9%

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

   if -1.3999999999999999 < y < -3.6e-40 or -1.84999999999999987e-51 < y < 9.00000000000000046e-115 or 1.01999999999999998e-81 < y < 7.4000000000000004e-56

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 81.1

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

   5. Applied rewrites 81.1%

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

   if -3.6e-40 < y < -1.84999999999999987e-51 or 9.00000000000000046e-115 < y < 1.01999999999999998e-81

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y \cdot z} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 75.4

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

   5. Applied rewrites 75.4%

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

   if 1.9000000000000001e264 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 50.8

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

   5. Applied rewrites 50.8%

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

   6. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.8%

      \[\leadsto y \cdot \color{blue}{x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4:\\ \;\;\;\;\left(z + x\right) \cdot y\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{elif}\;y \leq -1.85 \cdot 10^{-51}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-115}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{-81}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{-56}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+264}:\\ \;\;\;\;\left(z + x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 48.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{+72} \lor \neg \left(z \leq -8.6 \cdot 10^{-223} \lor \neg \left(z \leq -1.3 \cdot 10^{-236} \lor \neg \left(z \leq 4.6 \cdot 10^{-186} \lor \neg \left(z \leq 6.4 \cdot 10^{+307}\right)\right)\right)\right):\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -9.8e+72)
         (not
          (or (<= z -8.6e-223)
              (not
               (or (<= z -1.3e-236)
                   (not (or (<= z 4.6e-186) (not (<= z 6.4e+307)))))))))
   (* z y)
   (* y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -9.8e+72) || !((z <= -8.6e-223) || !((z <= -1.3e-236) || !((z <= 4.6e-186) || !(z <= 6.4e+307))))) {
		tmp = z * y;
	} 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 ((z <= (-9.8d+72)) .or. (.not. (z <= (-8.6d-223)) .or. (.not. (z <= (-1.3d-236)) .or. (.not. (z <= 4.6d-186) .or. (.not. (z <= 6.4d+307)))))) then
        tmp = z * y
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -9.8e+72) || !((z <= -8.6e-223) || !((z <= -1.3e-236) || !((z <= 4.6e-186) || !(z <= 6.4e+307))))) {
		tmp = z * y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -9.8e+72) or not ((z <= -8.6e-223) or not ((z <= -1.3e-236) or not ((z <= 4.6e-186) or not (z <= 6.4e+307)))):
		tmp = z * y
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -9.8e+72) || !((z <= -8.6e-223) || !((z <= -1.3e-236) || !((z <= 4.6e-186) || !(z <= 6.4e+307)))))
		tmp = Float64(z * y);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -9.8e+72) || ~(((z <= -8.6e-223) || ~(((z <= -1.3e-236) || ~(((z <= 4.6e-186) || ~((z <= 6.4e+307)))))))))
		tmp = z * y;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -9.8e+72], N[Not[Or[LessEqual[z, -8.6e-223], N[Not[Or[LessEqual[z, -1.3e-236], N[Not[Or[LessEqual[z, 4.6e-186], N[Not[LessEqual[z, 6.4e+307]], $MachinePrecision]]], $MachinePrecision]]], $MachinePrecision]]], $MachinePrecision]], N[(z * y), $MachinePrecision], N[(y * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.80000000000000012e72 or -8.5999999999999998e-223 < z < -1.3e-236 or 4.6000000000000002e-186 < z < 6.40000000000000011e307

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y \cdot z} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 60.3

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

   5. Applied rewrites 60.3%

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

   if -9.80000000000000012e72 < z < -8.5999999999999998e-223 or -1.3e-236 < z < 4.6000000000000002e-186 or 6.40000000000000011e307 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 91.6

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

   5. Applied rewrites 91.6%

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

   6. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.4%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{+72} \lor \neg \left(z \leq -8.6 \cdot 10^{-223} \lor \neg \left(z \leq -1.3 \cdot 10^{-236} \lor \neg \left(z \leq 4.6 \cdot 10^{-186} \lor \neg \left(z \leq 6.4 \cdot 10^{+307}\right)\right)\right)\right):\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{+82}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq 3800000:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+62}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+103}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+290}:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.1e+82)
   (* z y)
   (if (<= z 3800000.0)
     (fma y x x)
     (if (<= z 1.1e+62)
       (* z y)
       (if (<= z 3.5e+103) (fma y x x) (if (<= z 7e+290) (* z y) (* y x)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.1e+82) {
		tmp = z * y;
	} else if (z <= 3800000.0) {
		tmp = fma(y, x, x);
	} else if (z <= 1.1e+62) {
		tmp = z * y;
	} else if (z <= 3.5e+103) {
		tmp = fma(y, x, x);
	} else if (z <= 7e+290) {
		tmp = z * y;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.1e+82)
		tmp = Float64(z * y);
	elseif (z <= 3800000.0)
		tmp = fma(y, x, x);
	elseif (z <= 1.1e+62)
		tmp = Float64(z * y);
	elseif (z <= 3.5e+103)
		tmp = fma(y, x, x);
	elseif (z <= 7e+290)
		tmp = Float64(z * y);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.1e+82], N[(z * y), $MachinePrecision], If[LessEqual[z, 3800000.0], N[(y * x + x), $MachinePrecision], If[LessEqual[z, 1.1e+62], N[(z * y), $MachinePrecision], If[LessEqual[z, 3.5e+103], N[(y * x + x), $MachinePrecision], If[LessEqual[z, 7e+290], N[(z * y), $MachinePrecision], N[(y * x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.1000000000000003e82 or 3.8e6 < z < 1.10000000000000007e62 or 3.5e103 < z < 7.00000000000000026e290

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y \cdot z} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 74.2

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

   5. Applied rewrites 74.2%

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

   if -5.1000000000000003e82 < z < 3.8e6 or 1.10000000000000007e62 < z < 3.5e103

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 85.9

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

   5. Applied rewrites 85.9%

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

   if 7.00000000000000026e290 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 25.8

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

   5. Applied rewrites 25.8%

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

   6. Taylor expanded in y around inf

      \[\leadsto x \cdot \color{blue}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 1.2%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.1 \cdot 10^{+82}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq 3800000:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+62}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+103}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x\right)\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+290}:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 27.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return y * 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 = y * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. lower-fma.f64 62.2

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

5. Applied rewrites 62.2%

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

6. Taylor expanded in y around inf

   \[\leadsto x \cdot \color{blue}{y} \]
7. Step-by-step derivation

8. Applied rewrites 25.0%

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

9. Final simplification 25.0%

   \[\leadsto y \cdot x \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024338 
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))