Linear.Quaternion:$c/ from linear-1.19.1.3, A

Percentage Accurate: 98.1% → 98.7%

Time: 10.5s

Alternatives: 8

Speedup: 1.8×

• 34.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z, z \cdot 3, x \cdot y\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma z (* z 3.0) (* x y)))

C

double code(double x, double y, double z) {
	return fma(z, (z * 3.0), (x * y));
}

Julia

function code(x, y, z)
	return fma(z, Float64(z * 3.0), Float64(x * y))
end

Wolfram

code[x_, y_, z_] := N[(z * N[(z * 3.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.9%

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

   1. associate-+l+ N/A

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

   2. associate-+l+ N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-out N/A

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

   5. distribute-lft-out N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. +-lowering-+.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. *-lowering-*.f64 99.5

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

4. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, \left(z + z\right) + z, x \cdot y\right)} \]
5. Step-by-step derivation

   1. count-2 N/A

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

   2. distribute-lft1-in N/A

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

   3. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{3} \cdot z, x \cdot y\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot 3}, x \cdot y\right) \]

   5. *-lowering-*.f64 99.5

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot 3}, x \cdot y\right) \]

6. Applied egg-rr 99.5%

   \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot 3}, x \cdot y\right) \]
7. Add Preprocessing

Alternative 2: 84.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z + z, z, x \cdot y\right)\\ \mathbf{if}\;x \cdot y \leq -5.5 \cdot 10^{-244}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \cdot y \leq 4.3 \cdot 10^{-150}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (+ z z) z (* x y))))
   (if (<= (* x y) -5.5e-244)
     t_0
     (if (<= (* x y) 4.3e-150) (* z (* z 3.0)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((z + z), z, (x * y));
	double tmp;
	if ((x * y) <= -5.5e-244) {
		tmp = t_0;
	} else if ((x * y) <= 4.3e-150) {
		tmp = z * (z * 3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(z + z), z, Float64(x * y))
	tmp = 0.0
	if (Float64(x * y) <= -5.5e-244)
		tmp = t_0;
	elseif (Float64(x * y) <= 4.3e-150)
		tmp = Float64(z * Float64(z * 3.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z + z), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -5.5e-244], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 4.3e-150], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -5.4999999999999998e-244 or 4.30000000000000004e-150 < (*.f64 x y)

   1. Initial program 97.6%

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

      1. associate-+l+ N/A

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

      2. +-commutative N/A

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

      3. count-2 N/A

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

      4. associate-*r* N/A

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

      5. count-2 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 98.1

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

   4. Applied egg-rr 98.1%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 88.5

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

   7. Simplified 88.5%

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

   if -5.4999999999999998e-244 < (*.f64 x y) < 4.30000000000000004e-150

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + {z}^{2}} \]
   4. Step-by-step derivation

      1. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot {z}^{2}} \]

      2. metadata-eval N/A

         \[\leadsto \color{blue}{3} \cdot {z}^{2} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{{z}^{2} \cdot 3} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{z}^{2} \cdot 3} \]

      5. unpow2 N/A

         \[\leadsto \color{blue}{\left(z \cdot z\right)} \cdot 3 \]

      6. *-lowering-*.f64 90.5

         \[\leadsto \color{blue}{\left(z \cdot z\right)} \cdot 3 \]

   5. Simplified 90.5%

      \[\leadsto \color{blue}{\left(z \cdot z\right) \cdot 3} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{3 \cdot \left(z \cdot z\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(3 \cdot z\right) \cdot z} \]

      3. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{3}{1}} \cdot z\right) \cdot z \]

      4. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{3}{\frac{1}{z}}} \cdot z \]

      5. *-lft-identity N/A

         \[\leadsto \color{blue}{\left(1 \cdot \frac{3}{\frac{1}{z}}\right)} \cdot z \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 \cdot \frac{3}{\frac{1}{z}}\right) \cdot z} \]

      7. *-lft-identity N/A

         \[\leadsto \color{blue}{\frac{3}{\frac{1}{z}}} \cdot z \]

      8. associate-/r/ N/A

         \[\leadsto \color{blue}{\left(\frac{3}{1} \cdot z\right)} \cdot z \]

      9. metadata-eval N/A

         \[\leadsto \left(\color{blue}{3} \cdot z\right) \cdot z \]

      10. *-lowering-*.f64 90.8

          \[\leadsto \color{blue}{\left(3 \cdot z\right)} \cdot z \]

   7. Applied egg-rr 90.8%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5.5 \cdot 10^{-244}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, x \cdot y\right)\\ \mathbf{elif}\;x \cdot y \leq 4.3 \cdot 10^{-150}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+290}:\\ \;\;\;\;\mathsf{fma}\left(3, z \cdot z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 4e+290) (fma 3.0 (* z z) (* x y)) (fma z z (* x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4e+290) {
		tmp = fma(3.0, (z * z), (x * y));
	} else {
		tmp = fma(z, z, (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 4e+290)
		tmp = fma(3.0, Float64(z * z), Float64(x * y));
	else
		tmp = fma(z, z, Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 4e+290], N[(3.0 * N[(z * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(z * z + N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.00000000000000025e290

   1. Initial program 99.8%

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

      1. associate-+l+ N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. count-2 N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

         \[\leadsto \color{blue}{3} \cdot \left(z \cdot z\right) + x \cdot y \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(3, z \cdot z, x \cdot y\right)} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(3, \color{blue}{z \cdot z}, x \cdot y\right) \]

      9. *-lowering-*.f64 99.8

         \[\leadsto \mathsf{fma}\left(3, z \cdot z, \color{blue}{x \cdot y}\right) \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(3, z \cdot z, x \cdot y\right)} \]

   if 4.00000000000000025e290 < (*.f64 z z)

   1. Initial program 93.8%

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

      1. associate-+l+ N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-out N/A

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

      5. distribute-lft-out N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. *-lowering-*.f64 98.8

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

   4. Applied egg-rr 98.8%

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

   6. Applied egg-rr 98.8%

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{z}, x \cdot y\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 86.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(z, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e+68) (fma z z (* x y)) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+68) {
		tmp = fma(z, z, (x * y));
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+68)
		tmp = fma(z, z, Float64(x * y));
	else
		tmp = Float64(z * Float64(z * 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+68], N[(z * z + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000004e68

   1. Initial program 99.9%

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

      1. associate-+l+ N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-out N/A

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

      5. distribute-lft-out N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. *-lowering-*.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   6. Applied egg-rr 84.4%

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

   if 5.0000000000000004e68 < (*.f64 z z)

   1. Initial program 95.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + {z}^{2}} \]
   4. Step-by-step derivation

      1. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot {z}^{2}} \]

      2. metadata-eval N/A

         \[\leadsto \color{blue}{3} \cdot {z}^{2} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{{z}^{2} \cdot 3} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{z}^{2} \cdot 3} \]

      5. unpow2 N/A

         \[\leadsto \color{blue}{\left(z \cdot z\right)} \cdot 3 \]

      6. *-lowering-*.f64 90.4

         \[\leadsto \color{blue}{\left(z \cdot z\right)} \cdot 3 \]

   5. Simplified 90.4%

      \[\leadsto \color{blue}{\left(z \cdot z\right) \cdot 3} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{3 \cdot \left(z \cdot z\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(3 \cdot z\right) \cdot z} \]

      3. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{3}{1}} \cdot z\right) \cdot z \]

      4. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{3}{\frac{1}{z}}} \cdot z \]

      5. *-lft-identity N/A

         \[\leadsto \color{blue}{\left(1 \cdot \frac{3}{\frac{1}{z}}\right)} \cdot z \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 \cdot \frac{3}{\frac{1}{z}}\right) \cdot z} \]

      7. *-lft-identity N/A

         \[\leadsto \color{blue}{\frac{3}{\frac{1}{z}}} \cdot z \]

      8. associate-/r/ N/A

         \[\leadsto \color{blue}{\left(\frac{3}{1} \cdot z\right)} \cdot z \]

      9. metadata-eval N/A

         \[\leadsto \left(\color{blue}{3} \cdot z\right) \cdot z \]

      10. *-lowering-*.f64 90.5

          \[\leadsto \color{blue}{\left(3 \cdot z\right)} \cdot z \]

   7. Applied egg-rr 90.5%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(z, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+68}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e+68) (* x y) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+68) {
		tmp = x * y;
	} else {
		tmp = z * (z * 3.0);
	}
	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 * z) <= 5d+68) then
        tmp = x * y
    else
        tmp = z * (z * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+68) {
		tmp = x * y;
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 5e+68:
		tmp = x * y
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+68)
		tmp = Float64(x * y);
	else
		tmp = Float64(z * Float64(z * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 5e+68)
		tmp = x * y;
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+68], N[(x * y), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.0000000000000004e68

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 81.9

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

   5. Simplified 81.9%

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

   if 5.0000000000000004e68 < (*.f64 z z)

   1. Initial program 95.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot {z}^{2} + {z}^{2}} \]
   4. Step-by-step derivation

      1. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot {z}^{2}} \]

      2. metadata-eval N/A

         \[\leadsto \color{blue}{3} \cdot {z}^{2} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{{z}^{2} \cdot 3} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{z}^{2} \cdot 3} \]

      5. unpow2 N/A

         \[\leadsto \color{blue}{\left(z \cdot z\right)} \cdot 3 \]

      6. *-lowering-*.f64 90.4

         \[\leadsto \color{blue}{\left(z \cdot z\right)} \cdot 3 \]

   5. Simplified 90.4%

      \[\leadsto \color{blue}{\left(z \cdot z\right) \cdot 3} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{3 \cdot \left(z \cdot z\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(3 \cdot z\right) \cdot z} \]

      3. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{3}{1}} \cdot z\right) \cdot z \]

      4. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{3}{\frac{1}{z}}} \cdot z \]

      5. *-lft-identity N/A

         \[\leadsto \color{blue}{\left(1 \cdot \frac{3}{\frac{1}{z}}\right)} \cdot z \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 \cdot \frac{3}{\frac{1}{z}}\right) \cdot z} \]

      7. *-lft-identity N/A

         \[\leadsto \color{blue}{\frac{3}{\frac{1}{z}}} \cdot z \]

      8. associate-/r/ N/A

         \[\leadsto \color{blue}{\left(\frac{3}{1} \cdot z\right)} \cdot z \]

      9. metadata-eval N/A

         \[\leadsto \left(\color{blue}{3} \cdot z\right) \cdot z \]

      10. *-lowering-*.f64 90.5

          \[\leadsto \color{blue}{\left(3 \cdot z\right)} \cdot z \]

   7. Applied egg-rr 90.5%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+68}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2.35 \cdot 10^{+237}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2.35e+237) (* x y) (* z z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2.35e+237) {
		tmp = x * y;
	} else {
		tmp = z * 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 ((z * z) <= 2.35d+237) then
        tmp = x * y
    else
        tmp = z * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2.35e+237) {
		tmp = x * y;
	} else {
		tmp = z * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 2.35e+237:
		tmp = x * y
	else:
		tmp = z * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2.35e+237)
		tmp = Float64(x * y);
	else
		tmp = Float64(z * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2.35e+237)
		tmp = x * y;
	else
		tmp = z * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2.35e+237], N[(x * y), $MachinePrecision], N[(z * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.35000000000000009e237

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 72.3

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

   5. Simplified 72.3%

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

   if 2.35000000000000009e237 < (*.f64 z z)

   1. Initial program 94.1%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 89.4

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

   5. Simplified 89.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{z}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 90.6

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

   8. Simplified 90.6%

      \[\leadsto \color{blue}{z \cdot z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, x, 3 \cdot \left(z \cdot z\right)\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma y x (* 3.0 (* z z))))

C

double code(double x, double y, double z) {
	return fma(y, x, (3.0 * (z * z)));
}

Julia

function code(x, y, z)
	return fma(y, x, Float64(3.0 * Float64(z * z)))
end

Wolfram

code[x_, y_, z_] := N[(y * x + N[(3.0 * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.9%

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

   1. associate-+l+ N/A

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. count-2 N/A

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

   6. distribute-lft1-in N/A

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

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{3} \cdot \left(z \cdot z\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(y, x, \color{blue}{3 \cdot \left(z \cdot z\right)}\right) \]

   9. *-lowering-*.f64 98.3

      \[\leadsto \mathsf{fma}\left(y, x, 3 \cdot \color{blue}{\left(z \cdot z\right)}\right) \]

4. Applied egg-rr 98.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, 3 \cdot \left(z \cdot z\right)\right)} \]
5. Add Preprocessing

Alternative 8: 53.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

3. Taylor expanded in x around inf

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

   1. *-lowering-*.f64 53.0

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

5. Simplified 53.0%

   \[\leadsto \color{blue}{x \cdot y} \]
6. Add Preprocessing

Developer Target 1: 98.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot z\right) \cdot z + y \cdot x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* (* 3.0 z) z) (* y x)))

C

double code(double x, double y, double z) {
	return ((3.0 * z) * z) + (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 = ((3.0d0 * z) * z) + (y * x)
end function

Java

public static double code(double x, double y, double z) {
	return ((3.0 * z) * z) + (y * x);
}

Python

def code(x, y, z):
	return ((3.0 * z) * z) + (y * x)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(3.0 * z) * z) + Float64(y * x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((3.0 * z) * z) + (y * x);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(3.0 * z), $MachinePrecision] * z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024205 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* 3 z) z) (* y x)))

  (+ (+ (+ (* x y) (* z z)) (* z z)) (* z z)))