FastMath test5

Percentage Accurate: 99.9% → 100.0%

Time: 2.0min

Alternatives: 12

Speedup: 1.0×

• 44.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(d1 \cdot \left(\left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right) \cdot d1 \end{array} \]

FPCore

(FPCore (d1)
 :precision binary64
 (* (* d1 (* (* (* (* (* d1 (* d1 d1)) d1) d1) (* d1 d1)) d1)) d1))

C

double code(double d1) {
	return (d1 * (((((d1 * (d1 * d1)) * d1) * d1) * (d1 * d1)) * d1)) * d1;
}

Fortran

real(8) function code(d1)
    real(8), intent (in) :: d1
    code = (d1 * (((((d1 * (d1 * d1)) * d1) * d1) * (d1 * d1)) * d1)) * d1
end function

Java

public static double code(double d1) {
	return (d1 * (((((d1 * (d1 * d1)) * d1) * d1) * (d1 * d1)) * d1)) * d1;
}

Python

def code(d1):
	return (d1 * (((((d1 * (d1 * d1)) * d1) * d1) * (d1 * d1)) * d1)) * d1

Julia

function code(d1)
	return Float64(Float64(d1 * Float64(Float64(Float64(Float64(Float64(d1 * Float64(d1 * d1)) * d1) * d1) * Float64(d1 * d1)) * d1)) * d1)
end

MATLAB

function tmp = code(d1)
	tmp = (d1 * (((((d1 * (d1 * d1)) * d1) * d1) * (d1 * d1)) * d1)) * d1;
end

Wolfram

code[d1_] := N[(N[(d1 * N[(N[(N[(N[(N[(d1 * N[(d1 * d1), $MachinePrecision]), $MachinePrecision] * d1), $MachinePrecision] * d1), $MachinePrecision] * N[(d1 * d1), $MachinePrecision]), $MachinePrecision] * d1), $MachinePrecision]), $MachinePrecision] * d1), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(d1 \cdot \left(\left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right) \cdot d1 \end{array} \]

FPCore

(FPCore (d1)
 :precision binary64
 (* (* d1 (* (* (* (* (* d1 (* d1 d1)) d1) d1) (* d1 d1)) d1)) d1))

C

double code(double d1) {
	return (d1 * (((((d1 * (d1 * d1)) * d1) * d1) * (d1 * d1)) * d1)) * d1;
}

Fortran

real(8) function code(d1)
    real(8), intent (in) :: d1
    code = (d1 * (((((d1 * (d1 * d1)) * d1) * d1) * (d1 * d1)) * d1)) * d1
end function

Java

public static double code(double d1) {
	return (d1 * (((((d1 * (d1 * d1)) * d1) * d1) * (d1 * d1)) * d1)) * d1;
}

Python

def code(d1):
	return (d1 * (((((d1 * (d1 * d1)) * d1) * d1) * (d1 * d1)) * d1)) * d1

Julia

function code(d1)
	return Float64(Float64(d1 * Float64(Float64(Float64(Float64(Float64(d1 * Float64(d1 * d1)) * d1) * d1) * Float64(d1 * d1)) * d1)) * d1)
end

MATLAB

function tmp = code(d1)
	tmp = (d1 * (((((d1 * (d1 * d1)) * d1) * d1) * (d1 * d1)) * d1)) * d1;
end

Wolfram

code[d1_] := N[(N[(d1 * N[(N[(N[(N[(N[(d1 * N[(d1 * d1), $MachinePrecision]), $MachinePrecision] * d1), $MachinePrecision] * d1), $MachinePrecision] * N[(d1 * d1), $MachinePrecision]), $MachinePrecision] * d1), $MachinePrecision]), $MachinePrecision] * d1), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ {d1}^{10} \end{array} \]

FPCore

(FPCore (d1) :precision binary64 (pow d1 10.0))

C

double code(double d1) {
	return pow(d1, 10.0);
}

Fortran

real(8) function code(d1)
    real(8), intent (in) :: d1
    code = d1 ** 10.0d0
end function

Java

public static double code(double d1) {
	return Math.pow(d1, 10.0);
}

Python

def code(d1):
	return math.pow(d1, 10.0)

Julia

function code(d1)
	return d1 ^ 10.0
end

MATLAB

function tmp = code(d1)
	tmp = d1 ^ 10.0;
end

Wolfram

code[d1_] := N[Power[d1, 10.0], $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(d1 \cdot \left(\left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right) \cdot d1 \]
2. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto d1 \cdot \color{blue}{\left(d1 \cdot \left(\left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right)} \]

   2. associate-*r* N/A

      \[\leadsto \left(d1 \cdot d1\right) \cdot \color{blue}{\left(\left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)} \]

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

      \[\leadsto \mathsf{*.f64}\left(\left(d1 \cdot d1\right), \color{blue}{\left(\left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \left(\color{blue}{\left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)} \cdot d1\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot \color{blue}{\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right), \color{blue}{\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right), \left(\color{blue}{\left(d1 \cdot \left(d1 \cdot d1\right)\right)} \cdot d1\right)\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right), \left(\color{blue}{\left(d1 \cdot \left(d1 \cdot d1\right)\right)} \cdot d1\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(d1 \cdot d1\right), \left(d1 \cdot d1\right)\right), \left(\color{blue}{\left(d1 \cdot \left(d1 \cdot d1\right)\right)} \cdot d1\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \left(d1 \cdot d1\right)\right), \left(\left(\color{blue}{d1} \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(d1, d1\right)\right), \left(\left(d1 \cdot \color{blue}{\left(d1 \cdot d1\right)}\right) \cdot d1\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(d1, d1\right)\right), \left(d1 \cdot \color{blue}{\left(d1 \cdot \left(d1 \cdot d1\right)\right)}\right)\right)\right) \]

   14. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(d1, d1\right)\right), \left(\left(d1 \cdot d1\right) \cdot \color{blue}{\left(d1 \cdot d1\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(d1, d1\right)\right), \mathsf{*.f64}\left(\left(d1 \cdot d1\right), \color{blue}{\left(d1 \cdot d1\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(d1, d1\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \left(\color{blue}{d1} \cdot d1\right)\right)\right)\right) \]

   17. *-lowering-*.f64 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(d1, d1\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(d1, \color{blue}{d1}\right)\right)\right)\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left(d1 \cdot d1\right) \cdot \left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\left(d1 \cdot d1\right) \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot \color{blue}{\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)} \]

   2. unswap-sqr N/A

      \[\leadsto \left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right) \cdot \left(\color{blue}{\left(d1 \cdot d1\right)} \cdot \left(d1 \cdot d1\right)\right) \]

   3. pow2 N/A

      \[\leadsto {\left(d1 \cdot \left(d1 \cdot d1\right)\right)}^{2} \cdot \left(\color{blue}{\left(d1 \cdot d1\right)} \cdot \left(d1 \cdot d1\right)\right) \]

   4. pow2 N/A

      \[\leadsto {\left(d1 \cdot \left(d1 \cdot d1\right)\right)}^{2} \cdot {\left(d1 \cdot d1\right)}^{\color{blue}{2}} \]

   5. unpow-prod-down N/A

      \[\leadsto {\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(d1 \cdot d1\right)\right)}^{\color{blue}{2}} \]

   6. cube-unmult N/A

      \[\leadsto {\left({d1}^{3} \cdot \left(d1 \cdot d1\right)\right)}^{2} \]

   7. pow2 N/A

      \[\leadsto {\left({d1}^{3} \cdot {d1}^{2}\right)}^{2} \]

   8. pow-prod-up N/A

      \[\leadsto {\left({d1}^{\left(3 + 2\right)}\right)}^{2} \]

   9. pow-pow N/A

      \[\leadsto {d1}^{\color{blue}{\left(\left(3 + 2\right) \cdot 2\right)}} \]

   10. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{pow.f64}\left(d1, \color{blue}{\left(\left(3 + 2\right) \cdot 2\right)}\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{pow.f64}\left(d1, \left(5 \cdot 2\right)\right) \]

   12. metadata-eval 100.0%

       \[\leadsto \mathsf{pow.f64}\left(d1, 10\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{d1}^{10}} \]
7. Add Preprocessing

Alternative 2: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (d1)
 :precision binary64
 (* d1 (* d1 (* d1 (* d1 (* d1 (* d1 (* d1 (* d1 (* d1 d1))))))))))

C

double code(double d1) {
	return d1 * (d1 * (d1 * (d1 * (d1 * (d1 * (d1 * (d1 * (d1 * d1))))))));
}

Fortran

real(8) function code(d1)
    real(8), intent (in) :: d1
    code = d1 * (d1 * (d1 * (d1 * (d1 * (d1 * (d1 * (d1 * (d1 * d1))))))))
end function

Java

public static double code(double d1) {
	return d1 * (d1 * (d1 * (d1 * (d1 * (d1 * (d1 * (d1 * (d1 * d1))))))));
}

Python

def code(d1):
	return d1 * (d1 * (d1 * (d1 * (d1 * (d1 * (d1 * (d1 * (d1 * d1))))))))

Julia

function code(d1)
	return Float64(d1 * Float64(d1 * Float64(d1 * Float64(d1 * Float64(d1 * Float64(d1 * Float64(d1 * Float64(d1 * Float64(d1 * d1)))))))))
end

MATLAB

function tmp = code(d1)
	tmp = d1 * (d1 * (d1 * (d1 * (d1 * (d1 * (d1 * (d1 * (d1 * d1))))))));
end

Wolfram

code[d1_] := N[(d1 * N[(d1 * N[(d1 * N[(d1 * N[(d1 * N[(d1 * N[(d1 * N[(d1 * N[(d1 * d1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(d1 \cdot \left(\left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right) \cdot d1 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(d1 \cdot d1\right)\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right), d1\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   3. cube-unmult N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left({d1}^{3} \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   4. sqr-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot {d1}^{\left(\frac{3}{2}\right)}\right) \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right)\right), d1\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right)\right)\right), d1\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\left(\left(d1 \cdot d1\right) \cdot d1\right) \cdot d1\right) \cdot d1\right)\right)\right)\right), d1\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right)\right), d1\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot {d1}^{\left(\frac{3}{2}\right)}\right) \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right), d1\right) \]

   11. sqr-pow N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{3} \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right), d1\right) \]

   12. cube-unmult N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right), d1\right) \]

4. Applied egg-rr 99.9%

   \[\leadsto \left(d1 \cdot \color{blue}{\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)}\right) \cdot d1 \]
5. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right)\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right), d1\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right), d1\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(\left(d1 \cdot d1\right) \cdot d1\right)\right), d1\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right), d1\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot d1\right) \cdot d1\right) \cdot d1\right), d1\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot d1\right) \cdot d1\right), d1\right), d1\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right)\right)\right)\right) \cdot d1\right)} \cdot d1 \]

7. Final simplification 99.9%

   \[\leadsto d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right)\right)\right)\right)\right) \]
8. Add Preprocessing

Alternative 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\\ t\_0 \cdot t\_0 \end{array} \end{array} \]

FPCore

(FPCore (d1)
 :precision binary64
 (let* ((t_0 (* d1 (* d1 (* d1 (* d1 d1)))))) (* t_0 t_0)))

C

double code(double d1) {
	double t_0 = d1 * (d1 * (d1 * (d1 * d1)));
	return t_0 * t_0;
}

Fortran

real(8) function code(d1)
    real(8), intent (in) :: d1
    real(8) :: t_0
    t_0 = d1 * (d1 * (d1 * (d1 * d1)))
    code = t_0 * t_0
end function

Java

public static double code(double d1) {
	double t_0 = d1 * (d1 * (d1 * (d1 * d1)));
	return t_0 * t_0;
}

Python

def code(d1):
	t_0 = d1 * (d1 * (d1 * (d1 * d1)))
	return t_0 * t_0

Julia

function code(d1)
	t_0 = Float64(d1 * Float64(d1 * Float64(d1 * Float64(d1 * d1))))
	return Float64(t_0 * t_0)
end

MATLAB

function tmp = code(d1)
	t_0 = d1 * (d1 * (d1 * (d1 * d1)));
	tmp = t_0 * t_0;
end

Wolfram

code[d1_] := Block[{t$95$0 = N[(d1 * N[(d1 * N[(d1 * N[(d1 * d1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * t$95$0), $MachinePrecision]]

Derivation

1. Initial program 99.9%

   \[\left(d1 \cdot \left(\left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right) \cdot d1 \]
2. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto d1 \cdot \color{blue}{\left(d1 \cdot \left(\left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right)} \]

   2. associate-*r* N/A

      \[\leadsto \left(d1 \cdot d1\right) \cdot \color{blue}{\left(\left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)} \]

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

      \[\leadsto \mathsf{*.f64}\left(\left(d1 \cdot d1\right), \color{blue}{\left(\left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \left(\color{blue}{\left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)} \cdot d1\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot \color{blue}{\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right), \color{blue}{\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right), \left(\color{blue}{\left(d1 \cdot \left(d1 \cdot d1\right)\right)} \cdot d1\right)\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right), \left(\color{blue}{\left(d1 \cdot \left(d1 \cdot d1\right)\right)} \cdot d1\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(d1 \cdot d1\right), \left(d1 \cdot d1\right)\right), \left(\color{blue}{\left(d1 \cdot \left(d1 \cdot d1\right)\right)} \cdot d1\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \left(d1 \cdot d1\right)\right), \left(\left(\color{blue}{d1} \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(d1, d1\right)\right), \left(\left(d1 \cdot \color{blue}{\left(d1 \cdot d1\right)}\right) \cdot d1\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(d1, d1\right)\right), \left(d1 \cdot \color{blue}{\left(d1 \cdot \left(d1 \cdot d1\right)\right)}\right)\right)\right) \]

   14. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(d1, d1\right)\right), \left(\left(d1 \cdot d1\right) \cdot \color{blue}{\left(d1 \cdot d1\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(d1, d1\right)\right), \mathsf{*.f64}\left(\left(d1 \cdot d1\right), \color{blue}{\left(d1 \cdot d1\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(d1, d1\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \left(\color{blue}{d1} \cdot d1\right)\right)\right)\right) \]

   17. *-lowering-*.f64 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(d1, d1\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(d1, \color{blue}{d1}\right)\right)\right)\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left(d1 \cdot d1\right) \cdot \left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot \color{blue}{\left(d1 \cdot d1\right)} \]

   2. swap-sqr N/A

      \[\leadsto \left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot \color{blue}{\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)} \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(\left(\left(d1 \cdot d1\right) \cdot d1\right) \cdot d1\right) \cdot d1\right) \cdot \left(\left(\color{blue}{\left(d1 \cdot d1\right)} \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \]

   4. associate-*r* N/A

      \[\leadsto \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \left(\left(\left(\color{blue}{d1} \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \]

   5. associate-*r* N/A

      \[\leadsto \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \left(\left(\left(\left(d1 \cdot d1\right) \cdot d1\right) \cdot d1\right) \cdot d1\right) \]

   6. associate-*r* N/A

      \[\leadsto \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right), \color{blue}{\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)}\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(d1 \cdot d1\right)\right), \left(\color{blue}{\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)} \cdot d1\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(d1 \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right), \left(\color{blue}{\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)} \cdot d1\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right), \left(\color{blue}{\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)} \cdot d1\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right), \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \color{blue}{d1}\right) \cdot d1\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \mathsf{*.f64}\left(d1, \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right), \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \color{blue}{d1}\right) \cdot d1\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \mathsf{*.f64}\left(d1, \mathsf{*.f64}\left(d1, \left(d1 \cdot d1\right)\right)\right)\right), \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \mathsf{*.f64}\left(d1, \mathsf{*.f64}\left(d1, \mathsf{*.f64}\left(d1, d1\right)\right)\right)\right), \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right) \]

   15. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \mathsf{*.f64}\left(d1, \mathsf{*.f64}\left(d1, \mathsf{*.f64}\left(d1, d1\right)\right)\right)\right), \left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \color{blue}{\left(d1 \cdot d1\right)}\right)\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right)} \]
7. Add Preprocessing

Alternative 4: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\\ \left(d1 \cdot d1\right) \cdot \left(t\_0 \cdot t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (d1)
 :precision binary64
 (let* ((t_0 (* (* d1 d1) (* d1 d1)))) (* (* d1 d1) (* t_0 t_0))))

C

double code(double d1) {
	double t_0 = (d1 * d1) * (d1 * d1);
	return (d1 * d1) * (t_0 * t_0);
}

Fortran

real(8) function code(d1)
    real(8), intent (in) :: d1
    real(8) :: t_0
    t_0 = (d1 * d1) * (d1 * d1)
    code = (d1 * d1) * (t_0 * t_0)
end function

Java

public static double code(double d1) {
	double t_0 = (d1 * d1) * (d1 * d1);
	return (d1 * d1) * (t_0 * t_0);
}

Python

def code(d1):
	t_0 = (d1 * d1) * (d1 * d1)
	return (d1 * d1) * (t_0 * t_0)

Julia

function code(d1)
	t_0 = Float64(Float64(d1 * d1) * Float64(d1 * d1))
	return Float64(Float64(d1 * d1) * Float64(t_0 * t_0))
end

MATLAB

function tmp = code(d1)
	t_0 = (d1 * d1) * (d1 * d1);
	tmp = (d1 * d1) * (t_0 * t_0);
end

Wolfram

code[d1_] := Block[{t$95$0 = N[(N[(d1 * d1), $MachinePrecision] * N[(d1 * d1), $MachinePrecision]), $MachinePrecision]}, N[(N[(d1 * d1), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.9%

   \[\left(d1 \cdot \left(\left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right) \cdot d1 \]
2. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto d1 \cdot \color{blue}{\left(d1 \cdot \left(\left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right)} \]

   2. associate-*r* N/A

      \[\leadsto \left(d1 \cdot d1\right) \cdot \color{blue}{\left(\left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)} \]

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

      \[\leadsto \mathsf{*.f64}\left(\left(d1 \cdot d1\right), \color{blue}{\left(\left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \left(\color{blue}{\left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)} \cdot d1\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot \color{blue}{\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right), \color{blue}{\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right), \left(\color{blue}{\left(d1 \cdot \left(d1 \cdot d1\right)\right)} \cdot d1\right)\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right), \left(\color{blue}{\left(d1 \cdot \left(d1 \cdot d1\right)\right)} \cdot d1\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(d1 \cdot d1\right), \left(d1 \cdot d1\right)\right), \left(\color{blue}{\left(d1 \cdot \left(d1 \cdot d1\right)\right)} \cdot d1\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \left(d1 \cdot d1\right)\right), \left(\left(\color{blue}{d1} \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(d1, d1\right)\right), \left(\left(d1 \cdot \color{blue}{\left(d1 \cdot d1\right)}\right) \cdot d1\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(d1, d1\right)\right), \left(d1 \cdot \color{blue}{\left(d1 \cdot \left(d1 \cdot d1\right)\right)}\right)\right)\right) \]

   14. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(d1, d1\right)\right), \left(\left(d1 \cdot d1\right) \cdot \color{blue}{\left(d1 \cdot d1\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(d1, d1\right)\right), \mathsf{*.f64}\left(\left(d1 \cdot d1\right), \color{blue}{\left(d1 \cdot d1\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(d1, d1\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \left(\color{blue}{d1} \cdot d1\right)\right)\right)\right) \]

   17. *-lowering-*.f64 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(d1, d1\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, d1\right), \mathsf{*.f64}\left(d1, \color{blue}{d1}\right)\right)\right)\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left(d1 \cdot d1\right) \cdot \left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 5: 88.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (d1)
 :precision binary64
 (* d1 (* d1 (* d1 (* d1 (* d1 (* d1 (* d1 d1))))))))

C

double code(double d1) {
	return d1 * (d1 * (d1 * (d1 * (d1 * (d1 * (d1 * d1))))));
}

Fortran

real(8) function code(d1)
    real(8), intent (in) :: d1
    code = d1 * (d1 * (d1 * (d1 * (d1 * (d1 * (d1 * d1))))))
end function

Java

public static double code(double d1) {
	return d1 * (d1 * (d1 * (d1 * (d1 * (d1 * (d1 * d1))))));
}

Python

def code(d1):
	return d1 * (d1 * (d1 * (d1 * (d1 * (d1 * (d1 * d1))))))

Julia

function code(d1)
	return Float64(d1 * Float64(d1 * Float64(d1 * Float64(d1 * Float64(d1 * Float64(d1 * Float64(d1 * d1)))))))
end

MATLAB

function tmp = code(d1)
	tmp = d1 * (d1 * (d1 * (d1 * (d1 * (d1 * (d1 * d1))))));
end

Wolfram

code[d1_] := N[(d1 * N[(d1 * N[(d1 * N[(d1 * N[(d1 * N[(d1 * N[(d1 * d1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(d1 \cdot \left(\left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right) \cdot d1 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(d1 \cdot d1\right)\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right), d1\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   3. cube-unmult N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left({d1}^{3} \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   4. sqr-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot {d1}^{\left(\frac{3}{2}\right)}\right) \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right)\right), d1\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right)\right)\right), d1\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\left(\left(d1 \cdot d1\right) \cdot d1\right) \cdot d1\right) \cdot d1\right)\right)\right)\right), d1\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right)\right), d1\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot {d1}^{\left(\frac{3}{2}\right)}\right) \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right), d1\right) \]

   11. sqr-pow N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{3} \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right), d1\right) \]

   12. cube-unmult N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right), d1\right) \]

4. Applied egg-rr 99.9%

   \[\leadsto \left(d1 \cdot \color{blue}{\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)}\right) \cdot d1 \]
5. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right)\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right), d1\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right), d1\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(\left(d1 \cdot d1\right) \cdot d1\right)\right), d1\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right), d1\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot d1\right) \cdot d1\right) \cdot d1\right), d1\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot d1\right) \cdot d1\right), d1\right), d1\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right)\right)\right)\right) \cdot d1\right)} \cdot d1 \]

8. Applied egg-rr 89.1%

   \[\leadsto \left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \color{blue}{d1}\right)\right)\right)\right)\right) \cdot d1\right) \cdot d1 \]

9. Final simplification 89.1%

   \[\leadsto d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 6: 65.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (d1) :precision binary64 (* d1 (* d1 (* d1 (* d1 (* d1 (* d1 d1)))))))

C

double code(double d1) {
	return d1 * (d1 * (d1 * (d1 * (d1 * (d1 * d1)))));
}

Fortran

real(8) function code(d1)
    real(8), intent (in) :: d1
    code = d1 * (d1 * (d1 * (d1 * (d1 * (d1 * d1)))))
end function

Java

public static double code(double d1) {
	return d1 * (d1 * (d1 * (d1 * (d1 * (d1 * d1)))));
}

Python

def code(d1):
	return d1 * (d1 * (d1 * (d1 * (d1 * (d1 * d1)))))

Julia

function code(d1)
	return Float64(d1 * Float64(d1 * Float64(d1 * Float64(d1 * Float64(d1 * Float64(d1 * d1))))))
end

MATLAB

function tmp = code(d1)
	tmp = d1 * (d1 * (d1 * (d1 * (d1 * (d1 * d1)))));
end

Wolfram

code[d1_] := N[(d1 * N[(d1 * N[(d1 * N[(d1 * N[(d1 * N[(d1 * d1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(d1 \cdot \left(\left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right) \cdot d1 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(d1 \cdot d1\right)\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right), d1\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   3. cube-unmult N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left({d1}^{3} \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   4. sqr-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot {d1}^{\left(\frac{3}{2}\right)}\right) \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right)\right), d1\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right)\right)\right), d1\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\left(\left(d1 \cdot d1\right) \cdot d1\right) \cdot d1\right) \cdot d1\right)\right)\right)\right), d1\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right)\right), d1\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot {d1}^{\left(\frac{3}{2}\right)}\right) \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right), d1\right) \]

   11. sqr-pow N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{3} \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right), d1\right) \]

   12. cube-unmult N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right), d1\right) \]

4. Applied egg-rr 99.9%

   \[\leadsto \left(d1 \cdot \color{blue}{\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)}\right) \cdot d1 \]
5. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right)\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right), d1\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right), d1\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(\left(d1 \cdot d1\right) \cdot d1\right)\right), d1\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right), d1\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot d1\right) \cdot d1\right) \cdot d1\right), d1\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot d1\right) \cdot d1\right), d1\right), d1\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right)\right)\right)\right) \cdot d1\right)} \cdot d1 \]

8. Applied egg-rr 67.2%

   \[\leadsto \left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \color{blue}{d1}\right)\right)\right)\right) \cdot d1\right) \cdot d1 \]

9. Final simplification 67.2%

   \[\leadsto d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 7: 84.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (d1) :precision binary64 (* d1 (* d1 (* d1 (* d1 (* d1 d1))))))

C

double code(double d1) {
	return d1 * (d1 * (d1 * (d1 * (d1 * d1))));
}

Fortran

real(8) function code(d1)
    real(8), intent (in) :: d1
    code = d1 * (d1 * (d1 * (d1 * (d1 * d1))))
end function

Java

public static double code(double d1) {
	return d1 * (d1 * (d1 * (d1 * (d1 * d1))));
}

Python

def code(d1):
	return d1 * (d1 * (d1 * (d1 * (d1 * d1))))

Julia

function code(d1)
	return Float64(d1 * Float64(d1 * Float64(d1 * Float64(d1 * Float64(d1 * d1)))))
end

MATLAB

function tmp = code(d1)
	tmp = d1 * (d1 * (d1 * (d1 * (d1 * d1))));
end

Wolfram

code[d1_] := N[(d1 * N[(d1 * N[(d1 * N[(d1 * N[(d1 * d1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(d1 \cdot \left(\left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right) \cdot d1 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(d1 \cdot d1\right)\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right), d1\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   3. cube-unmult N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left({d1}^{3} \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   4. sqr-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot {d1}^{\left(\frac{3}{2}\right)}\right) \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right)\right), d1\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right)\right)\right), d1\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\left(\left(d1 \cdot d1\right) \cdot d1\right) \cdot d1\right) \cdot d1\right)\right)\right)\right), d1\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right)\right), d1\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot {d1}^{\left(\frac{3}{2}\right)}\right) \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right), d1\right) \]

   11. sqr-pow N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{3} \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right), d1\right) \]

   12. cube-unmult N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right), d1\right) \]

4. Applied egg-rr 99.9%

   \[\leadsto \left(d1 \cdot \color{blue}{\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)}\right) \cdot d1 \]
5. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right)\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right), d1\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right), d1\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(\left(d1 \cdot d1\right) \cdot d1\right)\right), d1\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right), d1\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot d1\right) \cdot d1\right) \cdot d1\right), d1\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot d1\right) \cdot d1\right), d1\right), d1\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right)\right)\right)\right) \cdot d1\right)} \cdot d1 \]

8. Applied egg-rr 84.7%

   \[\leadsto \left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \color{blue}{d1}\right)\right)\right) \cdot d1\right) \cdot d1 \]

9. Final simplification 84.7%

   \[\leadsto d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 8: 61.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right) \end{array} \]

FPCore

(FPCore (d1) :precision binary64 (* d1 (* d1 (* d1 (* d1 d1)))))

C

double code(double d1) {
	return d1 * (d1 * (d1 * (d1 * d1)));
}

Fortran

real(8) function code(d1)
    real(8), intent (in) :: d1
    code = d1 * (d1 * (d1 * (d1 * d1)))
end function

Java

public static double code(double d1) {
	return d1 * (d1 * (d1 * (d1 * d1)));
}

Python

def code(d1):
	return d1 * (d1 * (d1 * (d1 * d1)))

Julia

function code(d1)
	return Float64(d1 * Float64(d1 * Float64(d1 * Float64(d1 * d1))))
end

MATLAB

function tmp = code(d1)
	tmp = d1 * (d1 * (d1 * (d1 * d1)));
end

Wolfram

code[d1_] := N[(d1 * N[(d1 * N[(d1 * N[(d1 * d1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(d1 \cdot \left(\left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right) \cdot d1 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(d1 \cdot d1\right)\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right), d1\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   3. cube-unmult N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left({d1}^{3} \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   4. sqr-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot {d1}^{\left(\frac{3}{2}\right)}\right) \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right)\right), d1\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right)\right)\right), d1\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\left(\left(d1 \cdot d1\right) \cdot d1\right) \cdot d1\right) \cdot d1\right)\right)\right)\right), d1\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right)\right), d1\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot {d1}^{\left(\frac{3}{2}\right)}\right) \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right), d1\right) \]

   11. sqr-pow N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{3} \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right), d1\right) \]

   12. cube-unmult N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right), d1\right) \]

4. Applied egg-rr 99.9%

   \[\leadsto \left(d1 \cdot \color{blue}{\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)}\right) \cdot d1 \]
5. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right)\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right), d1\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right), d1\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(\left(d1 \cdot d1\right) \cdot d1\right)\right), d1\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right), d1\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot d1\right) \cdot d1\right) \cdot d1\right), d1\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot d1\right) \cdot d1\right), d1\right), d1\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right)\right)\right)\right) \cdot d1\right)} \cdot d1 \]

8. Applied egg-rr 62.1%

   \[\leadsto \left(\left(d1 \cdot \left(d1 \cdot \color{blue}{d1}\right)\right) \cdot d1\right) \cdot d1 \]

9. Final simplification 62.1%

   \[\leadsto d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right) \]
10. Add Preprocessing

Alternative 9: 76.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right) \end{array} \]

FPCore

(FPCore (d1) :precision binary64 (* d1 (* d1 (* d1 d1))))

C

double code(double d1) {
	return d1 * (d1 * (d1 * d1));
}

Fortran

real(8) function code(d1)
    real(8), intent (in) :: d1
    code = d1 * (d1 * (d1 * d1))
end function

Java

public static double code(double d1) {
	return d1 * (d1 * (d1 * d1));
}

Python

def code(d1):
	return d1 * (d1 * (d1 * d1))

Julia

function code(d1)
	return Float64(d1 * Float64(d1 * Float64(d1 * d1)))
end

MATLAB

function tmp = code(d1)
	tmp = d1 * (d1 * (d1 * d1));
end

Wolfram

code[d1_] := N[(d1 * N[(d1 * N[(d1 * d1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(d1 \cdot \left(\left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right) \cdot d1 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(d1 \cdot d1\right)\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right), d1\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   3. cube-unmult N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left({d1}^{3} \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   4. sqr-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot {d1}^{\left(\frac{3}{2}\right)}\right) \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right)\right), d1\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right)\right)\right), d1\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\left(\left(d1 \cdot d1\right) \cdot d1\right) \cdot d1\right) \cdot d1\right)\right)\right)\right), d1\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right)\right), d1\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot {d1}^{\left(\frac{3}{2}\right)}\right) \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right), d1\right) \]

   11. sqr-pow N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{3} \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right), d1\right) \]

   12. cube-unmult N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right), d1\right) \]

4. Applied egg-rr 99.9%

   \[\leadsto \left(d1 \cdot \color{blue}{\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)}\right) \cdot d1 \]
5. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right)\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right), d1\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right), d1\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(\left(d1 \cdot d1\right) \cdot d1\right)\right), d1\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right), d1\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot d1\right) \cdot d1\right) \cdot d1\right), d1\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot d1\right) \cdot d1\right), d1\right), d1\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right)\right)\right)\right) \cdot d1\right)} \cdot d1 \]

8. Applied egg-rr 78.8%

   \[\leadsto \left(\left(d1 \cdot \color{blue}{d1}\right) \cdot d1\right) \cdot d1 \]

9. Final simplification 78.8%

   \[\leadsto d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right) \]
10. Add Preprocessing

Alternative 10: 51.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ d1 \cdot \left(d1 \cdot d1\right) \end{array} \]

FPCore

(FPCore (d1) :precision binary64 (* d1 (* d1 d1)))

C

double code(double d1) {
	return d1 * (d1 * d1);
}

Fortran

real(8) function code(d1)
    real(8), intent (in) :: d1
    code = d1 * (d1 * d1)
end function

Java

public static double code(double d1) {
	return d1 * (d1 * d1);
}

Python

def code(d1):
	return d1 * (d1 * d1)

Julia

function code(d1)
	return Float64(d1 * Float64(d1 * d1))
end

MATLAB

function tmp = code(d1)
	tmp = d1 * (d1 * d1);
end

Wolfram

code[d1_] := N[(d1 * N[(d1 * d1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(d1 \cdot \left(\left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right) \cdot d1 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(d1 \cdot d1\right)\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right), d1\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   3. cube-unmult N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left({d1}^{3} \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   4. sqr-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot {d1}^{\left(\frac{3}{2}\right)}\right) \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right)\right), d1\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right)\right)\right), d1\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\left(\left(d1 \cdot d1\right) \cdot d1\right) \cdot d1\right) \cdot d1\right)\right)\right)\right), d1\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right)\right), d1\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot {d1}^{\left(\frac{3}{2}\right)}\right) \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right), d1\right) \]

   11. sqr-pow N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{3} \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right), d1\right) \]

   12. cube-unmult N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right), d1\right) \]

4. Applied egg-rr 99.9%

   \[\leadsto \left(d1 \cdot \color{blue}{\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)}\right) \cdot d1 \]
5. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right)\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right), d1\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right), d1\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(\left(d1 \cdot d1\right) \cdot d1\right)\right), d1\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right), d1\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot d1\right) \cdot d1\right) \cdot d1\right), d1\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot d1\right) \cdot d1\right), d1\right), d1\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right)\right)\right)\right) \cdot d1\right)} \cdot d1 \]

8. Applied egg-rr 51.7%

   \[\leadsto \left(\color{blue}{d1} \cdot d1\right) \cdot d1 \]

9. Final simplification 51.7%

   \[\leadsto d1 \cdot \left(d1 \cdot d1\right) \]
10. Add Preprocessing

Alternative 11: 52.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ d1 \cdot d1 \end{array} \]

FPCore

(FPCore (d1) :precision binary64 (* d1 d1))

C

double code(double d1) {
	return d1 * d1;
}

Fortran

real(8) function code(d1)
    real(8), intent (in) :: d1
    code = d1 * d1
end function

Java

public static double code(double d1) {
	return d1 * d1;
}

Python

def code(d1):
	return d1 * d1

Julia

function code(d1)
	return Float64(d1 * d1)
end

MATLAB

function tmp = code(d1)
	tmp = d1 * d1;
end

Wolfram

code[d1_] := N[(d1 * d1), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left(d1 \cdot \left(\left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right) \cdot d1 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(d1 \cdot d1\right)\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right), d1\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   3. cube-unmult N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left({d1}^{3} \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   4. sqr-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot {d1}^{\left(\frac{3}{2}\right)}\right) \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right)\right), d1\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right)\right)\right), d1\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\left(\left(d1 \cdot d1\right) \cdot d1\right) \cdot d1\right) \cdot d1\right)\right)\right)\right), d1\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right)\right), d1\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot {d1}^{\left(\frac{3}{2}\right)}\right) \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right), d1\right) \]

   11. sqr-pow N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{3} \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right), d1\right) \]

   12. cube-unmult N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right), d1\right) \]

4. Applied egg-rr 99.9%

   \[\leadsto \left(d1 \cdot \color{blue}{\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)}\right) \cdot d1 \]
5. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right)\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right), d1\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right), d1\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(\left(d1 \cdot d1\right) \cdot d1\right)\right), d1\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right), d1\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot d1\right) \cdot d1\right) \cdot d1\right), d1\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot d1\right) \cdot d1\right), d1\right), d1\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right)\right)\right)\right) \cdot d1\right)} \cdot d1 \]

8. Applied egg-rr 50.2%

   \[\leadsto \color{blue}{d1} \cdot d1 \]
9. Add Preprocessing

Alternative 12: 4.2% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ d1 \end{array} \]

FPCore

(FPCore (d1) :precision binary64 d1)

C

double code(double d1) {
	return d1;
}

Fortran

real(8) function code(d1)
    real(8), intent (in) :: d1
    code = d1
end function

Java

public static double code(double d1) {
	return d1;
}

Python

def code(d1):
	return d1

Julia

function code(d1)
	return d1
end

MATLAB

function tmp = code(d1)
	tmp = d1;
end

Wolfram

code[d1_] := d1

Derivation

1. Initial program 99.9%

   \[\left(d1 \cdot \left(\left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right) \cdot d1 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(d1 \cdot d1\right)\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right), d1\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   3. cube-unmult N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left({d1}^{3} \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   4. sqr-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot {d1}^{\left(\frac{3}{2}\right)}\right) \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot d1\right)\right), d1\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right)\right) \cdot d1\right)\right)\right), d1\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right)\right)\right)\right), d1\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\left(\left(d1 \cdot d1\right) \cdot d1\right) \cdot d1\right) \cdot d1\right)\right)\right)\right), d1\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left({d1}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right)\right), d1\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left({d1}^{\left(\frac{3}{2}\right)} \cdot {d1}^{\left(\frac{3}{2}\right)}\right) \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right), d1\right) \]

   11. sqr-pow N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left({d1}^{3} \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right), d1\right) \]

   12. cube-unmult N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(d1, \left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot \left(\left(\left(d1 \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right) \cdot d1\right)\right)\right), d1\right) \]

4. Applied egg-rr 99.9%

   \[\leadsto \left(d1 \cdot \color{blue}{\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)}\right) \cdot d1 \]
5. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right)\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right), d1\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right), d1\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(\left(d1 \cdot d1\right) \cdot d1\right)\right), d1\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot \left(d1 \cdot d1\right)\right) \cdot d1\right), d1\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot d1\right) \cdot d1\right) \cdot d1\right), d1\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(\left(d1 \cdot d1\right) \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right) \cdot d1\right) \cdot d1\right), d1\right), d1\right) \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\left(\left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot \left(d1 \cdot d1\right)\right)\right)\right)\right)\right)\right) \cdot d1\right)} \cdot d1 \]

8. Applied egg-rr 4.3%

   \[\leadsto \color{blue}{d1} \]
9. Add Preprocessing

Developer Target 1: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ {d1}^{10} \end{array} \]

FPCore

(FPCore (d1) :precision binary64 (pow d1 10.0))

C

double code(double d1) {
	return pow(d1, 10.0);
}

Fortran

real(8) function code(d1)
    real(8), intent (in) :: d1
    code = d1 ** 10.0d0
end function

Java

public static double code(double d1) {
	return Math.pow(d1, 10.0);
}

Python

def code(d1):
	return math.pow(d1, 10.0)

Julia

function code(d1)
	return d1 ^ 10.0
end

MATLAB

function tmp = code(d1)
	tmp = d1 ^ 10.0;
end

Wolfram

code[d1_] := N[Power[d1, 10.0], $MachinePrecision]

Reproduce

herbie shell --seed 2024138 
(FPCore (d1)
  :name "FastMath test5"
  :precision binary64

  :alt
  (! :herbie-platform default (pow d1 10))

  (* (* d1 (* (* (* (* (* d1 (* d1 d1)) d1) d1) (* d1 d1)) d1)) d1))