From Rump in a 1983 paper

Percentage Accurate: 18.8% → 100.0%

Time: 7.4s

Alternatives: 4

Speedup: 9.3×

Specification

\[x = 10864 \land y = 18817\]

Math

\[\begin{array}{l} \\ \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (- (* 9.0 (pow x 4.0)) (pow y 4.0)) (* 2.0 (* y y))))

C

double code(double x, double y) {
	return ((9.0 * pow(x, 4.0)) - pow(y, 4.0)) + (2.0 * (y * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((9.0d0 * (x ** 4.0d0)) - (y ** 4.0d0)) + (2.0d0 * (y * y))
end function

Java

public static double code(double x, double y) {
	return ((9.0 * Math.pow(x, 4.0)) - Math.pow(y, 4.0)) + (2.0 * (y * y));
}

Python

def code(x, y):
	return ((9.0 * math.pow(x, 4.0)) - math.pow(y, 4.0)) + (2.0 * (y * y))

Julia

function code(x, y)
	return Float64(Float64(Float64(9.0 * (x ^ 4.0)) - (y ^ 4.0)) + Float64(2.0 * Float64(y * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = ((9.0 * (x ^ 4.0)) - (y ^ 4.0)) + (2.0 * (y * y));
end

Wolfram

code[x_, y_] := N[(N[(N[(9.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] - N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 18.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (- (* 9.0 (pow x 4.0)) (pow y 4.0)) (* 2.0 (* y y))))

C

double code(double x, double y) {
	return ((9.0 * pow(x, 4.0)) - pow(y, 4.0)) + (2.0 * (y * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((9.0d0 * (x ** 4.0d0)) - (y ** 4.0d0)) + (2.0d0 * (y * y))
end function

Java

public static double code(double x, double y) {
	return ((9.0 * Math.pow(x, 4.0)) - Math.pow(y, 4.0)) + (2.0 * (y * y));
}

Python

def code(x, y):
	return ((9.0 * math.pow(x, 4.0)) - math.pow(y, 4.0)) + (2.0 * (y * y))

Julia

function code(x, y)
	return Float64(Float64(Float64(9.0 * (x ^ 4.0)) - (y ^ 4.0)) + Float64(2.0 * Float64(y * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = ((9.0 * (x ^ 4.0)) - (y ^ 4.0)) + (2.0 * (y * y));
end

Wolfram

code[x_, y_] := N[(N[(N[(9.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] - N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(2 - y \cdot y, y \cdot y, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 9\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (fma (- 2.0 (* y y)) (* y y) (* (* (* x x) (* x x)) 9.0)))

C

double code(double x, double y) {
	return fma((2.0 - (y * y)), (y * y), (((x * x) * (x * x)) * 9.0));
}

Julia

function code(x, y)
	return fma(Float64(2.0 - Float64(y * y)), Float64(y * y), Float64(Float64(Float64(x * x) * Float64(x * x)) * 9.0))
end

Wolfram

code[x_, y_] := N[(N[(2.0 - N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.8%

   \[\left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(9 \cdot {x}^{4} - {y}^{4}\right)}, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]
4. Step-by-step derivation

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(9 \cdot {x}^{4}\right), \left({y}^{4}\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(9, \left({x}^{4}\right)\right), \left({y}^{4}\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(9, \mathsf{pow.f64}\left(x, 4\right)\right), \left({y}^{4}\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(9, \mathsf{pow.f64}\left(x, 4\right)\right), \left({y}^{\left(3 + 1\right)}\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(9, \mathsf{pow.f64}\left(x, 4\right)\right), \left({y}^{\left(\left(2 + 1\right) + 1\right)}\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   6. pow-plus N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(9, \mathsf{pow.f64}\left(x, 4\right)\right), \left({y}^{\left(2 + 1\right)} \cdot y\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   7. pow-plus N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(9, \mathsf{pow.f64}\left(x, 4\right)\right), \left(\left({y}^{2} \cdot y\right) \cdot y\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(9, \mathsf{pow.f64}\left(x, 4\right)\right), \left(y \cdot \left({y}^{2} \cdot y\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(9, \mathsf{pow.f64}\left(x, 4\right)\right), \mathsf{*.f64}\left(y, \left({y}^{2} \cdot y\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(9, \mathsf{pow.f64}\left(x, 4\right)\right), \mathsf{*.f64}\left(y, \left(y \cdot {y}^{2}\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(9, \mathsf{pow.f64}\left(x, 4\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2}\right)\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(9, \mathsf{pow.f64}\left(x, 4\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot y\right)\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   13. *-lowering-*.f64 18.8%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(9, \mathsf{pow.f64}\left(x, 4\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

5. Simplified 18.8%

   \[\leadsto \color{blue}{\left(9 \cdot {x}^{4} - y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)} + 2 \cdot \left(y \cdot y\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto 2 \cdot \left(y \cdot y\right) + \color{blue}{\left(9 \cdot {x}^{4} - y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto 2 \cdot \left(y \cdot y\right) + \left(9 \cdot {x}^{4} + \color{blue}{\left(\mathsf{neg}\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)}\right) \]

   3. +-commutative N/A

      \[\leadsto 2 \cdot \left(y \cdot y\right) + \left(\left(\mathsf{neg}\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right) + \color{blue}{9 \cdot {x}^{4}}\right) \]

   4. associate-+r+ N/A

      \[\leadsto \left(2 \cdot \left(y \cdot y\right) + \left(\mathsf{neg}\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right) + \color{blue}{9 \cdot {x}^{4}} \]

   5. unsub-neg N/A

      \[\leadsto \left(2 \cdot \left(y \cdot y\right) - y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) + \color{blue}{9} \cdot {x}^{4} \]

   6. associate-*r* N/A

      \[\leadsto \left(2 \cdot \left(y \cdot y\right) - \left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) + 9 \cdot {x}^{4} \]

   7. distribute-rgt-out-- N/A

      \[\leadsto \left(y \cdot y\right) \cdot \left(2 - y \cdot y\right) + \color{blue}{9} \cdot {x}^{4} \]

   8. *-commutative N/A

      \[\leadsto \left(2 - y \cdot y\right) \cdot \left(y \cdot y\right) + \color{blue}{9} \cdot {x}^{4} \]

   9. fma-define N/A

      \[\leadsto \mathsf{fma}\left(2 - y \cdot y, \color{blue}{y \cdot y}, 9 \cdot {x}^{4}\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\left(2 - y \cdot y\right), \color{blue}{\left(y \cdot y\right)}, \left(9 \cdot {x}^{4}\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \left(y \cdot y\right)\right), \left(\color{blue}{y} \cdot y\right), \left(9 \cdot {x}^{4}\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \left(y \cdot y\right), \left(9 \cdot {x}^{4}\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, \color{blue}{y}\right), \left(9 \cdot {x}^{4}\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(9, \left({x}^{4}\right)\right)\right) \]

   15. rem-exp-log N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(9, \left({\left(e^{\log x}\right)}^{4}\right)\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(9, \mathsf{pow.f64}\left(\left(e^{\log x}\right), 4\right)\right)\right) \]

   17. rem-exp-log 100.0%

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(9, \mathsf{pow.f64}\left(x, 4\right)\right)\right) \]

7. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2 - y \cdot y, y \cdot y, 9 \cdot {x}^{4}\right)} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \left({x}^{4} \cdot 9\right)\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\left({x}^{4}\right), 9\right)\right) \]

   3. pow-to-exp N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\left(e^{\log x \cdot 4}\right), 9\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\left(e^{4 \cdot \log x}\right), 9\right)\right) \]

   5. exp-prod N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\left({\left(e^{4}\right)}^{\log x}\right), 9\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\left({\left(e^{2 + 2}\right)}^{\log x}\right), 9\right)\right) \]

   7. prod-exp N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\left({\left(e^{2} \cdot e^{2}\right)}^{\log x}\right), 9\right)\right) \]

   8. unpow-prod-down N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\left({\left(e^{2}\right)}^{\log x} \cdot {\left(e^{2}\right)}^{\log x}\right), 9\right)\right) \]

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

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\left(e^{2}\right)}^{\log x}\right), \left({\left(e^{2}\right)}^{\log x}\right)\right), 9\right)\right) \]

   10. pow-exp N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{2 \cdot \log x}\right), \left({\left(e^{2}\right)}^{\log x}\right)\right), 9\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{\log x \cdot 2}\right), \left({\left(e^{2}\right)}^{\log x}\right)\right), 9\right)\right) \]

   12. exp-lft-sqr N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{\log x} \cdot e^{\log x}\right), \left({\left(e^{2}\right)}^{\log x}\right)\right), 9\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(e^{\log x}\right), \left(e^{\log x}\right)\right), \left({\left(e^{2}\right)}^{\log x}\right)\right), 9\right)\right) \]

   14. rem-exp-log N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(e^{\log x}\right)\right), \left({\left(e^{2}\right)}^{\log x}\right)\right), 9\right)\right) \]

   15. rem-exp-log N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({\left(e^{2}\right)}^{\log x}\right)\right), 9\right)\right) \]

   16. pow-exp N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(e^{2 \cdot \log x}\right)\right), 9\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(e^{\log x \cdot 2}\right)\right), 9\right)\right) \]

   18. exp-lft-sqr N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(e^{\log x} \cdot e^{\log x}\right)\right), 9\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(e^{\log x}\right), \left(e^{\log x}\right)\right)\right), 9\right)\right) \]

   20. rem-exp-log N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \left(e^{\log x}\right)\right)\right), 9\right)\right) \]

   21. rem-exp-log 100.0%

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right), 9\right)\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto \mathsf{fma}\left(2 - y \cdot y, y \cdot y, \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 9}\right) \]
10. Add Preprocessing

Alternative 2: 18.8% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 9\right) - y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - y \cdot \left(y \cdot -2\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (- (- (* (* x x) (* (* x x) 9.0)) (* y (* y (* y y)))) (* y (* y -2.0))))

C

double code(double x, double y) {
	return (((x * x) * ((x * x) * 9.0)) - (y * (y * (y * y)))) - (y * (y * -2.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (((x * x) * ((x * x) * 9.0d0)) - (y * (y * (y * y)))) - (y * (y * (-2.0d0)))
end function

Java

public static double code(double x, double y) {
	return (((x * x) * ((x * x) * 9.0)) - (y * (y * (y * y)))) - (y * (y * -2.0));
}

Python

def code(x, y):
	return (((x * x) * ((x * x) * 9.0)) - (y * (y * (y * y)))) - (y * (y * -2.0))

Julia

function code(x, y)
	return Float64(Float64(Float64(Float64(x * x) * Float64(Float64(x * x) * 9.0)) - Float64(y * Float64(y * Float64(y * y)))) - Float64(y * Float64(y * -2.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (((x * x) * ((x * x) * 9.0)) - (y * (y * (y * y)))) - (y * (y * -2.0));
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision] - N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * N[(y * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.8%

   \[\left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(9 \cdot {x}^{4} - {y}^{4}\right)}, \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]
4. Step-by-step derivation

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\left(9 \cdot {x}^{4}\right), \left({y}^{4}\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(9, \left({x}^{4}\right)\right), \left({y}^{4}\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(9, \mathsf{pow.f64}\left(x, 4\right)\right), \left({y}^{4}\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(9, \mathsf{pow.f64}\left(x, 4\right)\right), \left({y}^{\left(3 + 1\right)}\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(9, \mathsf{pow.f64}\left(x, 4\right)\right), \left({y}^{\left(\left(2 + 1\right) + 1\right)}\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   6. pow-plus N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(9, \mathsf{pow.f64}\left(x, 4\right)\right), \left({y}^{\left(2 + 1\right)} \cdot y\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   7. pow-plus N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(9, \mathsf{pow.f64}\left(x, 4\right)\right), \left(\left({y}^{2} \cdot y\right) \cdot y\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(9, \mathsf{pow.f64}\left(x, 4\right)\right), \left(y \cdot \left({y}^{2} \cdot y\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(9, \mathsf{pow.f64}\left(x, 4\right)\right), \mathsf{*.f64}\left(y, \left({y}^{2} \cdot y\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(9, \mathsf{pow.f64}\left(x, 4\right)\right), \mathsf{*.f64}\left(y, \left(y \cdot {y}^{2}\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(9, \mathsf{pow.f64}\left(x, 4\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2}\right)\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(9, \mathsf{pow.f64}\left(x, 4\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot y\right)\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   13. *-lowering-*.f64 18.8%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(9, \mathsf{pow.f64}\left(x, 4\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

5. Simplified 18.8%

   \[\leadsto \color{blue}{\left(9 \cdot {x}^{4} - y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)} + 2 \cdot \left(y \cdot y\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto 2 \cdot \left(y \cdot y\right) + \color{blue}{\left(9 \cdot {x}^{4} - y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto 2 \cdot \left(y \cdot y\right) + \left(9 \cdot {x}^{4} + \color{blue}{\left(\mathsf{neg}\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)}\right) \]

   3. +-commutative N/A

      \[\leadsto 2 \cdot \left(y \cdot y\right) + \left(\left(\mathsf{neg}\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right) + \color{blue}{9 \cdot {x}^{4}}\right) \]

   4. associate-+r+ N/A

      \[\leadsto \left(2 \cdot \left(y \cdot y\right) + \left(\mathsf{neg}\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right) + \color{blue}{9 \cdot {x}^{4}} \]

   5. unsub-neg N/A

      \[\leadsto \left(2 \cdot \left(y \cdot y\right) - y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) + \color{blue}{9} \cdot {x}^{4} \]

   6. associate-*r* N/A

      \[\leadsto \left(2 \cdot \left(y \cdot y\right) - \left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) + 9 \cdot {x}^{4} \]

   7. distribute-rgt-out-- N/A

      \[\leadsto \left(y \cdot y\right) \cdot \left(2 - y \cdot y\right) + \color{blue}{9} \cdot {x}^{4} \]

   8. *-commutative N/A

      \[\leadsto \left(2 - y \cdot y\right) \cdot \left(y \cdot y\right) + \color{blue}{9} \cdot {x}^{4} \]

   9. fma-define N/A

      \[\leadsto \mathsf{fma}\left(2 - y \cdot y, \color{blue}{y \cdot y}, 9 \cdot {x}^{4}\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\left(2 - y \cdot y\right), \color{blue}{\left(y \cdot y\right)}, \left(9 \cdot {x}^{4}\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \left(y \cdot y\right)\right), \left(\color{blue}{y} \cdot y\right), \left(9 \cdot {x}^{4}\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \left(y \cdot y\right), \left(9 \cdot {x}^{4}\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, \color{blue}{y}\right), \left(9 \cdot {x}^{4}\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(9, \left({x}^{4}\right)\right)\right) \]

   15. rem-exp-log N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(9, \left({\left(e^{\log x}\right)}^{4}\right)\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(9, \mathsf{pow.f64}\left(\left(e^{\log x}\right), 4\right)\right)\right) \]

   17. rem-exp-log 100.0%

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(9, \mathsf{pow.f64}\left(x, 4\right)\right)\right) \]

7. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2 - y \cdot y, y \cdot y, 9 \cdot {x}^{4}\right)} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(y \cdot y\right) \cdot \left(2 - y \cdot y\right) + \color{blue}{9} \cdot {x}^{4} \]

   2. sub-neg N/A

      \[\leadsto \left(y \cdot y\right) \cdot \left(2 + \left(\mathsf{neg}\left(y \cdot y\right)\right)\right) + 9 \cdot {x}^{4} \]

   3. distribute-lft-in N/A

      \[\leadsto \left(\left(y \cdot y\right) \cdot 2 + \left(y \cdot y\right) \cdot \left(\mathsf{neg}\left(y \cdot y\right)\right)\right) + \color{blue}{9} \cdot {x}^{4} \]

   4. associate-*r* N/A

      \[\leadsto \left(y \cdot \left(y \cdot 2\right) + \left(y \cdot y\right) \cdot \left(\mathsf{neg}\left(y \cdot y\right)\right)\right) + 9 \cdot {x}^{4} \]

   5. associate-+l+ N/A

      \[\leadsto y \cdot \left(y \cdot 2\right) + \color{blue}{\left(\left(y \cdot y\right) \cdot \left(\mathsf{neg}\left(y \cdot y\right)\right) + 9 \cdot {x}^{4}\right)} \]

   6. +-commutative N/A

      \[\leadsto \left(\left(y \cdot y\right) \cdot \left(\mathsf{neg}\left(y \cdot y\right)\right) + 9 \cdot {x}^{4}\right) + \color{blue}{y \cdot \left(y \cdot 2\right)} \]

   7. cancel-sign-sub N/A

      \[\leadsto \left(\left(y \cdot y\right) \cdot \left(\mathsf{neg}\left(y \cdot y\right)\right) + 9 \cdot {x}^{4}\right) - \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(y \cdot 2\right)} \]

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

      \[\leadsto \mathsf{\_.f64}\left(\left(\left(y \cdot y\right) \cdot \left(\mathsf{neg}\left(y \cdot y\right)\right) + 9 \cdot {x}^{4}\right), \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \left(y \cdot 2\right)\right)}\right) \]

9. Applied egg-rr 18.8%

   \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 9\right) - y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - y \cdot \left(y \cdot -2\right)} \]
10. Add Preprocessing

Alternative 3: 11.1% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \left(y \cdot y\right) \cdot \left(2 + y \cdot y\right) - y \cdot \left(y \cdot \left(y \cdot y\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (- (* (* y y) (+ 2.0 (* y y))) (* y (* y (* y y)))))

C

double code(double x, double y) {
	return ((y * y) * (2.0 + (y * y))) - (y * (y * (y * y)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((y * y) * (2.0d0 + (y * y))) - (y * (y * (y * y)))
end function

Java

public static double code(double x, double y) {
	return ((y * y) * (2.0 + (y * y))) - (y * (y * (y * y)));
}

Python

def code(x, y):
	return ((y * y) * (2.0 + (y * y))) - (y * (y * (y * y)))

Julia

function code(x, y)
	return Float64(Float64(Float64(y * y) * Float64(2.0 + Float64(y * y))) - Float64(y * Float64(y * Float64(y * y))))
end

MATLAB

function tmp = code(x, y)
	tmp = ((y * y) * (2.0 + (y * y))) - (y * (y * (y * y)));
end

Wolfram

code[x_, y_] := N[(N[(N[(y * y), $MachinePrecision] * N[(2.0 + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.8%

   \[\left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. metadata-eval N/A

      \[\leadsto 2 \cdot {y}^{2} - {y}^{\left(3 + \color{blue}{1}\right)} \]

   2. metadata-eval N/A

      \[\leadsto 2 \cdot {y}^{2} - {y}^{\left(\left(2 + 1\right) + 1\right)} \]

   3. pow-plus N/A

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

   4. pow-plus N/A

      \[\leadsto 2 \cdot {y}^{2} - \left({y}^{2} \cdot y\right) \cdot y \]

   5. cancel-sign-sub-inv N/A

      \[\leadsto 2 \cdot {y}^{2} + \color{blue}{\left(\mathsf{neg}\left({y}^{2} \cdot y\right)\right) \cdot y} \]

   6. unpow2 N/A

      \[\leadsto 2 \cdot \left(y \cdot y\right) + \left(\mathsf{neg}\left({y}^{2} \cdot y\right)\right) \cdot y \]

   7. associate-*r* N/A

      \[\leadsto \left(2 \cdot y\right) \cdot y + \color{blue}{\left(\mathsf{neg}\left({y}^{2} \cdot y\right)\right)} \cdot y \]

   8. distribute-lft-neg-out N/A

      \[\leadsto \left(2 \cdot y\right) \cdot y + \left(\left(\mathsf{neg}\left({y}^{2}\right)\right) \cdot y\right) \cdot y \]

   9. mul-1-neg N/A

      \[\leadsto \left(2 \cdot y\right) \cdot y + \left(\left(-1 \cdot {y}^{2}\right) \cdot y\right) \cdot y \]

   10. distribute-rgt-out N/A

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

   11. distribute-rgt-in N/A

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

   12. *-commutative N/A

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

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

       \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\left(2 + -1 \cdot {y}^{2}\right) \cdot y\right)}\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left(2 + -1 \cdot {y}^{2}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(2 + -1 \cdot {y}^{2}\right)}\right)\right) \]

   16. mul-1-neg N/A

       \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(2 + \left(\mathsf{neg}\left({y}^{2}\right)\right)\right)\right)\right) \]

   17. unsub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(2 - \color{blue}{{y}^{2}}\right)\right)\right) \]

   18. --lowering--.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(2, \color{blue}{\left({y}^{2}\right)}\right)\right)\right) \]

   19. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(2, \left(y \cdot \color{blue}{y}\right)\right)\right)\right) \]

   20. *-lowering-*.f64 1.5%

       \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right) \]

5. Simplified 1.5%

   \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(2 - y \cdot y\right)\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{2}\right)\right) \]
7. Step-by-step derivation

8. Simplified 11.1%

   \[\leadsto y \cdot \left(y \cdot \color{blue}{2}\right) \]
9. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto y \cdot \left(y \cdot 2\right) + \color{blue}{0} \]

   2. +-inverses N/A

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

   3. sub-neg N/A

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

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

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

   5. +-commutative N/A

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

   6. associate-+r+ N/A

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

   7. associate-*r* N/A

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

   8. distribute-lft-in N/A

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

   9. sub-neg N/A

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

   10. *-commutative N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. distribute-lft-out-- N/A

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

   14. associate-*r* N/A

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

10. Applied egg-rr 11.1%

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

11. Final simplification 11.1%

    \[\leadsto \left(y \cdot y\right) \cdot \left(2 + y \cdot y\right) - y \cdot \left(y \cdot \left(y \cdot y\right)\right) \]
12. Add Preprocessing

Alternative 4: 11.1% accurate, 42.6× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(2 \cdot y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* y (* 2.0 y)))

C

double code(double x, double y) {
	return y * (2.0 * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y * (2.0d0 * y)
end function

Java

public static double code(double x, double y) {
	return y * (2.0 * y);
}

Python

def code(x, y):
	return y * (2.0 * y)

Julia

function code(x, y)
	return Float64(y * Float64(2.0 * y))
end

MATLAB

function tmp = code(x, y)
	tmp = y * (2.0 * y);
end

Wolfram

code[x_, y_] := N[(y * N[(2.0 * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.8%

   \[\left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. metadata-eval N/A

      \[\leadsto 2 \cdot {y}^{2} - {y}^{\left(3 + \color{blue}{1}\right)} \]

   2. metadata-eval N/A

      \[\leadsto 2 \cdot {y}^{2} - {y}^{\left(\left(2 + 1\right) + 1\right)} \]

   3. pow-plus N/A

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

   4. pow-plus N/A

      \[\leadsto 2 \cdot {y}^{2} - \left({y}^{2} \cdot y\right) \cdot y \]

   5. cancel-sign-sub-inv N/A

      \[\leadsto 2 \cdot {y}^{2} + \color{blue}{\left(\mathsf{neg}\left({y}^{2} \cdot y\right)\right) \cdot y} \]

   6. unpow2 N/A

      \[\leadsto 2 \cdot \left(y \cdot y\right) + \left(\mathsf{neg}\left({y}^{2} \cdot y\right)\right) \cdot y \]

   7. associate-*r* N/A

      \[\leadsto \left(2 \cdot y\right) \cdot y + \color{blue}{\left(\mathsf{neg}\left({y}^{2} \cdot y\right)\right)} \cdot y \]

   8. distribute-lft-neg-out N/A

      \[\leadsto \left(2 \cdot y\right) \cdot y + \left(\left(\mathsf{neg}\left({y}^{2}\right)\right) \cdot y\right) \cdot y \]

   9. mul-1-neg N/A

      \[\leadsto \left(2 \cdot y\right) \cdot y + \left(\left(-1 \cdot {y}^{2}\right) \cdot y\right) \cdot y \]

   10. distribute-rgt-out N/A

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

   11. distribute-rgt-in N/A

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

   12. *-commutative N/A

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

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

       \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\left(2 + -1 \cdot {y}^{2}\right) \cdot y\right)}\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left(2 + -1 \cdot {y}^{2}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(2 + -1 \cdot {y}^{2}\right)}\right)\right) \]

   16. mul-1-neg N/A

       \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(2 + \left(\mathsf{neg}\left({y}^{2}\right)\right)\right)\right)\right) \]

   17. unsub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(2 - \color{blue}{{y}^{2}}\right)\right)\right) \]

   18. --lowering--.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(2, \color{blue}{\left({y}^{2}\right)}\right)\right)\right) \]

   19. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(2, \left(y \cdot \color{blue}{y}\right)\right)\right)\right) \]

   20. *-lowering-*.f64 1.5%

       \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right) \]

5. Simplified 1.5%

   \[\leadsto \color{blue}{y \cdot \left(y \cdot \left(2 - y \cdot y\right)\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{2}\right)\right) \]
7. Step-by-step derivation

8. Simplified 11.1%

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

9. Final simplification 11.1%

   \[\leadsto y \cdot \left(2 \cdot y\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024164 
(FPCore (x y)
  :name "From Rump in a 1983 paper"
  :precision binary64
  :pre (and (== x 10864.0) (== y 18817.0))
  (+ (- (* 9.0 (pow x 4.0)) (pow y 4.0)) (* 2.0 (* y y))))