From Rump in a 1983 paper

Percentage Accurate: 18.8% → 100.0%

Time: 6.0s

Alternatives: 2

Speedup: 5.6×

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, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+l- N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-pow.f64 N/A

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

   7. metadata-eval N/A

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

   8. pow-pow N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 N/A

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

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

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

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

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

   15. lower-fma.f64 N/A

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

4. Applied rewrites 100.0%

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

Alternative 2: 11.1% accurate, 20.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(2.0 * N[(y * 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. sub-neg N/A

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

   2. +-commutative N/A

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

   3. neg-sub0 N/A

      \[\leadsto \color{blue}{\left(0 - {y}^{4}\right)} + 2 \cdot {y}^{2} \]

   4. associate-+l- N/A

      \[\leadsto \color{blue}{0 - \left({y}^{4} - 2 \cdot {y}^{2}\right)} \]

   5. sub0-neg N/A

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

   6. lower-neg.f64 N/A

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

   7. metadata-eval N/A

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

   8. pow-sqr N/A

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

   9. unpow2 N/A

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

   10. associate-*r* N/A

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

   11. unpow2 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

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

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

   15. lower-*.f64 N/A

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

   16. *-commutative N/A

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

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

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

   18. lower-*.f64 N/A

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

   19. sub-neg N/A

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

   20. unpow2 N/A

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

   21. lower-fma.f64 N/A

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

   22. metadata-eval 1.5

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

5. Applied rewrites 1.5%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 11.1%

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

Reproduce

herbie shell --seed 2024223 
(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))))