Main:bigenough2 from A

Percentage Accurate: 100.0% → 100.0%

Time: 3.9s

Alternatives: 8

Speedup: 1.0×

• 20.9% 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: 58.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+156}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq -4.3 \cdot 10^{+15}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{-32}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq -4.3 \cdot 10^{-113}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-140}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{-145}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+171}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.1e+156)
   (* y z)
   (if (<= y -4.3e+15)
     (* y x)
     (if (<= y -1.15e-32)
       (* y z)
       (if (<= y -4.3e-113)
         x
         (if (<= y -8.5e-140)
           (* y z)
           (if (<= y 5.3e-145) x (if (<= y 9.2e+171) (* y z) (* y x)))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.1e+156) {
		tmp = y * z;
	} else if (y <= -4.3e+15) {
		tmp = y * x;
	} else if (y <= -1.15e-32) {
		tmp = y * z;
	} else if (y <= -4.3e-113) {
		tmp = x;
	} else if (y <= -8.5e-140) {
		tmp = y * z;
	} else if (y <= 5.3e-145) {
		tmp = x;
	} else if (y <= 9.2e+171) {
		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 <= (-3.1d+156)) then
        tmp = y * z
    else if (y <= (-4.3d+15)) then
        tmp = y * x
    else if (y <= (-1.15d-32)) then
        tmp = y * z
    else if (y <= (-4.3d-113)) then
        tmp = x
    else if (y <= (-8.5d-140)) then
        tmp = y * z
    else if (y <= 5.3d-145) then
        tmp = x
    else if (y <= 9.2d+171) 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 <= -3.1e+156) {
		tmp = y * z;
	} else if (y <= -4.3e+15) {
		tmp = y * x;
	} else if (y <= -1.15e-32) {
		tmp = y * z;
	} else if (y <= -4.3e-113) {
		tmp = x;
	} else if (y <= -8.5e-140) {
		tmp = y * z;
	} else if (y <= 5.3e-145) {
		tmp = x;
	} else if (y <= 9.2e+171) {
		tmp = y * z;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.1e+156:
		tmp = y * z
	elif y <= -4.3e+15:
		tmp = y * x
	elif y <= -1.15e-32:
		tmp = y * z
	elif y <= -4.3e-113:
		tmp = x
	elif y <= -8.5e-140:
		tmp = y * z
	elif y <= 5.3e-145:
		tmp = x
	elif y <= 9.2e+171:
		tmp = y * z
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.1e+156)
		tmp = Float64(y * z);
	elseif (y <= -4.3e+15)
		tmp = Float64(y * x);
	elseif (y <= -1.15e-32)
		tmp = Float64(y * z);
	elseif (y <= -4.3e-113)
		tmp = x;
	elseif (y <= -8.5e-140)
		tmp = Float64(y * z);
	elseif (y <= 5.3e-145)
		tmp = x;
	elseif (y <= 9.2e+171)
		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 <= -3.1e+156)
		tmp = y * z;
	elseif (y <= -4.3e+15)
		tmp = y * x;
	elseif (y <= -1.15e-32)
		tmp = y * z;
	elseif (y <= -4.3e-113)
		tmp = x;
	elseif (y <= -8.5e-140)
		tmp = y * z;
	elseif (y <= 5.3e-145)
		tmp = x;
	elseif (y <= 9.2e+171)
		tmp = y * z;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.1e+156], N[(y * z), $MachinePrecision], If[LessEqual[y, -4.3e+15], N[(y * x), $MachinePrecision], If[LessEqual[y, -1.15e-32], N[(y * z), $MachinePrecision], If[LessEqual[y, -4.3e-113], x, If[LessEqual[y, -8.5e-140], N[(y * z), $MachinePrecision], If[LessEqual[y, 5.3e-145], x, If[LessEqual[y, 9.2e+171], N[(y * z), $MachinePrecision], N[(y * x), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.1000000000000002e156 or -4.3e15 < y < -1.15e-32 or -4.3e-113 < y < -8.49999999999999997e-140 or 5.29999999999999999e-145 < y < 9.20000000000000069e171

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 75.5%

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

   4. Taylor expanded in x around 0 66.5%

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

   if -3.1000000000000002e156 < y < -4.3e15 or 9.20000000000000069e171 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around -inf 93.0%

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

      1. +-commutative 93.0%

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

      2. mul-1-neg 93.0%

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

      3. unsub-neg 93.0%

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

      4. sub-neg 93.0%

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

      5. metadata-eval 93.0%

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

      6. +-commutative 93.0%

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

      7. mul-1-neg 93.0%

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

      8. unsub-neg 93.0%

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

   5. Simplified 93.0%

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

   6. Taylor expanded in y around inf 100.0%

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

      1. neg-mul-1 100.0%

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

      2. distribute-lft-out-- 92.9%

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

      3. distribute-rgt-neg-in 92.9%

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

      4. distribute-lft-neg-out 92.9%

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

      5. cancel-sign-sub 92.9%

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

      6. distribute-lft-in 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in z around 0 69.6%

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

       1. *-commutative 69.6%

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

   11. Simplified 69.6%

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

   if -1.15e-32 < y < -4.3e-113 or -8.49999999999999997e-140 < y < 5.29999999999999999e-145

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 78.7%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+156}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq -4.3 \cdot 10^{+15}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{-32}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq -4.3 \cdot 10^{-113}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-140}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{-145}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+171}:\\ \;\;\;\;y \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x + z\right)\\ \mathbf{if}\;y \leq -1.4 \cdot 10^{-32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{-113}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-140}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-145}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ x z))))
   (if (<= y -1.4e-32)
     t_0
     (if (<= y -4.5e-113)
       x
       (if (<= y -9e-140) (* y z) (if (<= y 2.3e-145) x t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x + z);
	double tmp;
	if (y <= -1.4e-32) {
		tmp = t_0;
	} else if (y <= -4.5e-113) {
		tmp = x;
	} else if (y <= -9e-140) {
		tmp = y * z;
	} else if (y <= 2.3e-145) {
		tmp = x;
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = y * (x + z)
    if (y <= (-1.4d-32)) then
        tmp = t_0
    else if (y <= (-4.5d-113)) then
        tmp = x
    else if (y <= (-9d-140)) then
        tmp = y * z
    else if (y <= 2.3d-145) then
        tmp = x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (x + z);
	double tmp;
	if (y <= -1.4e-32) {
		tmp = t_0;
	} else if (y <= -4.5e-113) {
		tmp = x;
	} else if (y <= -9e-140) {
		tmp = y * z;
	} else if (y <= 2.3e-145) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (x + z)
	tmp = 0
	if y <= -1.4e-32:
		tmp = t_0
	elif y <= -4.5e-113:
		tmp = x
	elif y <= -9e-140:
		tmp = y * z
	elif y <= 2.3e-145:
		tmp = x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(x + z))
	tmp = 0.0
	if (y <= -1.4e-32)
		tmp = t_0;
	elseif (y <= -4.5e-113)
		tmp = x;
	elseif (y <= -9e-140)
		tmp = Float64(y * z);
	elseif (y <= 2.3e-145)
		tmp = x;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (x + z);
	tmp = 0.0;
	if (y <= -1.4e-32)
		tmp = t_0;
	elseif (y <= -4.5e-113)
		tmp = x;
	elseif (y <= -9e-140)
		tmp = y * z;
	elseif (y <= 2.3e-145)
		tmp = x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(x + z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.4e-32], t$95$0, If[LessEqual[y, -4.5e-113], x, If[LessEqual[y, -9e-140], N[(y * z), $MachinePrecision], If[LessEqual[y, 2.3e-145], x, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.3999999999999999e-32 or 2.30000000000000007e-145 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around -inf 96.3%

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

      1. +-commutative 96.3%

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

      2. mul-1-neg 96.3%

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

      3. unsub-neg 96.3%

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

      4. sub-neg 96.3%

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

      5. metadata-eval 96.3%

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

      6. +-commutative 96.3%

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

      7. mul-1-neg 96.3%

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

      8. unsub-neg 96.3%

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

   5. Simplified 96.3%

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

   6. Taylor expanded in y around inf 94.3%

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

      1. neg-mul-1 94.3%

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

      2. distribute-lft-out-- 90.7%

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

      3. distribute-rgt-neg-in 90.7%

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

      4. distribute-lft-neg-out 90.7%

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

      5. cancel-sign-sub 90.7%

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

      6. distribute-lft-in 94.3%

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

   8. Simplified 94.3%

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

   if -1.3999999999999999e-32 < y < -4.5000000000000001e-113 or -9.00000000000000008e-140 < y < 2.30000000000000007e-145

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 78.7%

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

   if -4.5000000000000001e-113 < y < -9.00000000000000008e-140

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 100.0%

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

   4. Taylor expanded in x around 0 85.9%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-32}:\\ \;\;\;\;y \cdot \left(x + z\right)\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{-113}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-140}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-145}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x + z\right)\\ t_1 := x + y \cdot x\\ \mathbf{if}\;y \leq -1.25 \cdot 10^{-32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-113}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-140}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{-145}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (+ x z))) (t_1 (+ x (* y x))))
   (if (<= y -1.25e-32)
     t_0
     (if (<= y -5.8e-113)
       t_1
       (if (<= y -9e-140) (* y z) (if (<= y 5.3e-145) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x + z);
	double t_1 = x + (y * x);
	double tmp;
	if (y <= -1.25e-32) {
		tmp = t_0;
	} else if (y <= -5.8e-113) {
		tmp = t_1;
	} else if (y <= -9e-140) {
		tmp = y * z;
	} else if (y <= 5.3e-145) {
		tmp = t_1;
	} else {
		tmp = t_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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y * (x + z)
    t_1 = x + (y * x)
    if (y <= (-1.25d-32)) then
        tmp = t_0
    else if (y <= (-5.8d-113)) then
        tmp = t_1
    else if (y <= (-9d-140)) then
        tmp = y * z
    else if (y <= 5.3d-145) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (x + z);
	double t_1 = x + (y * x);
	double tmp;
	if (y <= -1.25e-32) {
		tmp = t_0;
	} else if (y <= -5.8e-113) {
		tmp = t_1;
	} else if (y <= -9e-140) {
		tmp = y * z;
	} else if (y <= 5.3e-145) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (x + z)
	t_1 = x + (y * x)
	tmp = 0
	if y <= -1.25e-32:
		tmp = t_0
	elif y <= -5.8e-113:
		tmp = t_1
	elif y <= -9e-140:
		tmp = y * z
	elif y <= 5.3e-145:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(x + z))
	t_1 = Float64(x + Float64(y * x))
	tmp = 0.0
	if (y <= -1.25e-32)
		tmp = t_0;
	elseif (y <= -5.8e-113)
		tmp = t_1;
	elseif (y <= -9e-140)
		tmp = Float64(y * z);
	elseif (y <= 5.3e-145)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (x + z);
	t_1 = x + (y * x);
	tmp = 0.0;
	if (y <= -1.25e-32)
		tmp = t_0;
	elseif (y <= -5.8e-113)
		tmp = t_1;
	elseif (y <= -9e-140)
		tmp = y * z;
	elseif (y <= 5.3e-145)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(x + z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.25e-32], t$95$0, If[LessEqual[y, -5.8e-113], t$95$1, If[LessEqual[y, -9e-140], N[(y * z), $MachinePrecision], If[LessEqual[y, 5.3e-145], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.25e-32 or 5.29999999999999999e-145 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around -inf 96.3%

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

      1. +-commutative 96.3%

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

      2. mul-1-neg 96.3%

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

      3. unsub-neg 96.3%

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

      4. sub-neg 96.3%

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

      5. metadata-eval 96.3%

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

      6. +-commutative 96.3%

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

      7. mul-1-neg 96.3%

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

      8. unsub-neg 96.3%

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

   5. Simplified 96.3%

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

   6. Taylor expanded in y around inf 94.3%

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

      1. neg-mul-1 94.3%

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

      2. distribute-lft-out-- 90.7%

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

      3. distribute-rgt-neg-in 90.7%

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

      4. distribute-lft-neg-out 90.7%

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

      5. cancel-sign-sub 90.7%

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

      6. distribute-lft-in 94.3%

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

   8. Simplified 94.3%

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

   if -1.25e-32 < y < -5.80000000000000008e-113 or -9.00000000000000008e-140 < y < 5.29999999999999999e-145

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0 78.7%

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

      1. *-commutative 3.3%

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

   5. Simplified 78.7%

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

   if -5.80000000000000008e-113 < y < -9.00000000000000008e-140

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 100.0%

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

   4. Taylor expanded in x around 0 85.9%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-32}:\\ \;\;\;\;y \cdot \left(x + z\right)\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-113}:\\ \;\;\;\;x + y \cdot x\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-140}:\\ \;\;\;\;y \cdot z\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{-145}:\\ \;\;\;\;x + y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -20500000000000 \lor \neg \left(y \leq 9.5 \cdot 10^{-17}\right):\\ \;\;\;\;y \cdot \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -20500000000000.0) (not (<= y 9.5e-17)))
   (* y (+ x z))
   (+ x (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -20500000000000.0) || !(y <= 9.5e-17)) {
		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 <= (-20500000000000.0d0)) .or. (.not. (y <= 9.5d-17))) 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 <= -20500000000000.0) || !(y <= 9.5e-17)) {
		tmp = y * (x + z);
	} else {
		tmp = x + (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -20500000000000.0) or not (y <= 9.5e-17):
		tmp = y * (x + z)
	else:
		tmp = x + (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -20500000000000.0) || !(y <= 9.5e-17))
		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 <= -20500000000000.0) || ~((y <= 9.5e-17)))
		tmp = y * (x + z);
	else
		tmp = x + (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -20500000000000.0], N[Not[LessEqual[y, 9.5e-17]], $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 < -2.05e13 or 9.50000000000000029e-17 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around -inf 95.3%

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

      1. +-commutative 95.3%

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

      2. mul-1-neg 95.3%

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

      3. unsub-neg 95.3%

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

      4. sub-neg 95.3%

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

      5. metadata-eval 95.3%

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

      6. +-commutative 95.3%

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

      7. mul-1-neg 95.3%

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

      8. unsub-neg 95.3%

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

   5. Simplified 95.3%

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

   6. Taylor expanded in y around inf 99.4%

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

      1. neg-mul-1 99.4%

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

      2. distribute-lft-out-- 94.7%

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

      3. distribute-rgt-neg-in 94.7%

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

      4. distribute-lft-neg-out 94.7%

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

      5. cancel-sign-sub 94.7%

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

      6. distribute-lft-in 99.4%

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

   8. Simplified 99.4%

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

   if -2.05e13 < y < 9.50000000000000029e-17

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -20500000000000 \lor \neg \left(y \leq 9.5 \cdot 10^{-17}\right):\\ \;\;\;\;y \cdot \left(x + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-10} \lor \neg \left(y \leq 4.2 \cdot 10^{-9}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4.8e-10) (not (<= y 4.2e-9))) (* y x) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.8e-10) || !(y <= 4.2e-9)) {
		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 <= (-4.8d-10)) .or. (.not. (y <= 4.2d-9))) 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 <= -4.8e-10) || !(y <= 4.2e-9)) {
		tmp = y * x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4.8e-10) or not (y <= 4.2e-9):
		tmp = y * x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4.8e-10) || !(y <= 4.2e-9))
		tmp = Float64(y * x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -4.8e-10) || ~((y <= 4.2e-9)))
		tmp = y * x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4.8e-10], N[Not[LessEqual[y, 4.2e-9]], $MachinePrecision]], N[(y * x), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -4.8e-10 or 4.20000000000000039e-9 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around -inf 95.4%

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

      1. +-commutative 95.4%

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

      2. mul-1-neg 95.4%

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

      3. unsub-neg 95.4%

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

      4. sub-neg 95.4%

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

      5. metadata-eval 95.4%

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

      6. +-commutative 95.4%

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

      7. mul-1-neg 95.4%

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

      8. unsub-neg 95.4%

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

   5. Simplified 95.4%

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

   6. Taylor expanded in y around inf 99.4%

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

      1. neg-mul-1 99.4%

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

      2. distribute-lft-out-- 94.9%

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

      3. distribute-rgt-neg-in 94.9%

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

      4. distribute-lft-neg-out 94.9%

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

      5. cancel-sign-sub 94.9%

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

      6. distribute-lft-in 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in z around 0 51.5%

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

       1. *-commutative 51.5%

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

   11. Simplified 51.5%

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

   if -4.8e-10 < y < 4.20000000000000039e-9

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 62.8%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-10} \lor \neg \left(y \leq 4.2 \cdot 10^{-9}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
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: 36.6% 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 32.0%

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

4. Final simplification 32.0%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024043 
(FPCore (x y z)
  :name "Main:bigenough2 from A"
  :precision binary64
  (+ x (* y (+ z x))))