From Rump in a 1983 paper, rewritten

Percentage Accurate: 3.1% → 100.0%

Time: 4.5s

Alternatives: 2

Speedup: 11.6×

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, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[((-y) * y + 2.0), $MachinePrecision] * N[(y * y), $MachinePrecision] + N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * 9.0), $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. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

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

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

   7. lower-fma.f64 N/A

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

   8. lift--.f64 N/A

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

   9. sub-neg N/A

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

   10. distribute-neg-in N/A

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

   11. lift-*.f64 N/A

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

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

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-neg.f64 100.0

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-pow N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. pow2 N/A

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

   7. lower-*.f64 100.0

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

6. Applied rewrites 100.0%

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

Alternative 2: 11.1% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(y * y), $MachinePrecision] * 2.0), $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 y around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{9 \cdot {x}^{4} + 2 \cdot {y}^{2}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{{x}^{4} \cdot 9} + 2 \cdot {y}^{2} \]

   3. metadata-eval N/A

      \[\leadsto {x}^{4} \cdot 9 + \color{blue}{\left(-2 \cdot -1\right)} \cdot {y}^{2} \]

   4. associate-*r* N/A

      \[\leadsto {x}^{4} \cdot 9 + \color{blue}{-2 \cdot \left(-1 \cdot {y}^{2}\right)} \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, 9, -2 \cdot \left(-1 \cdot {y}^{2}\right)\right)} \]

   6. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{4}}, 9, -2 \cdot \left(-1 \cdot {y}^{2}\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left({x}^{4}, 9, \color{blue}{\left(-2 \cdot -1\right) \cdot {y}^{2}}\right) \]

   8. metadata-eval N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 9.6

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

5. Applied rewrites 9.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 11.1%

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

Reproduce

herbie shell --seed 2024314 
(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))))