From Rump in a 1983 paper, rewritten

Percentage Accurate: 3.1% → 100.0%

Time: 4.7s

Alternatives: 3

Speedup: 22.8×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 3.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 3.1%

   \[9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

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

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

   5. fma-define N/A

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

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

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

   7. neg-sub0 N/A

      \[\leadsto \mathsf{fma.f64}\left(\left(0 - \left(y \cdot y - 2\right)\right), \left(\color{blue}{y} \cdot y\right), \left(9 \cdot {x}^{4}\right)\right) \]

   8. associate-+l- N/A

      \[\leadsto \mathsf{fma.f64}\left(\left(\left(0 - y \cdot y\right) + 2\right), \left(\color{blue}{y} \cdot y\right), \left(9 \cdot {x}^{4}\right)\right) \]

   9. neg-sub0 N/A

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

   10. +-commutative N/A

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

   11. sub-neg N/A

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

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

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

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

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

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

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

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

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

   16. rem-exp-log N/A

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

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

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

   18. rem-exp-log 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 2: 18.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return ((9.0 * pow(x, 4.0)) - (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 = ((9.0d0 * (x ** 4.0d0)) - (y * (y * (y * y)))) - (y * (y * (-2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(N[(9.0 * N[Power[x, 4.0], $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 3.1%

   \[9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

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

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

   2. sub-neg N/A

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

   3. distribute-rgt-in N/A

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

   4. associate-+r+ N/A

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

   5. *-commutative N/A

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

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

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

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

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

4. Applied egg-rr 18.8%

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

Alternative 3: 11.1% accurate, 22.8× 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 3.1%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-1 \cdot \left({y}^{2} \cdot \left({y}^{2} - 2\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

   5. *-commutative N/A

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

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

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

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

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

   8. *-commutative N/A

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

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

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

   10. neg-sub0 N/A

       \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(0 - \color{blue}{\left({y}^{2} - 2\right)}\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(0 - \left({y}^{2} + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(0 - \left({y}^{2} + -2\right)\right)\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(0 - \left(-2 + \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

   14. associate--r+ N/A

       \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\left(0 - -2\right) - \color{blue}{{y}^{2}}\right)\right)\right) \]

   15. metadata-eval N/A

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

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

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

   17. unpow2 N/A

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

   18. *-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, rewritten"
  :precision binary64
  :pre (and (== x 10864.0) (== y 18817.0))
  (- (* 9.0 (pow x 4.0)) (* (* y y) (- (* y y) 2.0))))