From Rump in a 1983 paper, rewritten

Percentage Accurate: 3.1% → 100.0%

Time: 3.4s

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

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

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

Wolfram

code[x_, y_] := (-N[(N[(N[(y * y + -2.0), $MachinePrecision] * y), $MachinePrecision] * y + N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -9.0), $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. Taylor expanded in x around 0

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

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

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

   2. mul-1-neg N/A

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

   3. associate-*r* N/A

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

   4. +-commutative N/A

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

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

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

   6. *-commutative N/A

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

   7. fp-cancel-sub-sign-inv N/A

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

   8. mul-1-neg N/A

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

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

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

   10. *-commutative N/A

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

   11. distribute-neg-out N/A

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

   12. lower-neg.f64 N/A

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

   13. *-commutative N/A

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

   14. unpow2 N/A

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

   15. associate-*r* N/A

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

   16. lower-fma.f64 N/A

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 100.0%

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

Alternative 2: 11.1% accurate, 11.6× speedup

Math

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

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(Float64(2.0 * y) * y)
end

MATLAB

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

Wolfram

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

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

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

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

   6. lower-*.f64 N/A

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

5. Applied rewrites 1.5%

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

6. Taylor expanded in y around 0

   \[\leadsto \left(2 \cdot y\right) \cdot y \]
7. Step-by-step derivation

8. Applied rewrites 11.1%

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

Reproduce

herbie shell --seed 2024332 
(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))))