From Rump in a 1983 paper, rewritten

Percentage Accurate: 3.1% → 100.0%

Time: 5.1s

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(y * N[(y * N[(2.0 - N[(y * y), $MachinePrecision]), $MachinePrecision]), $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. 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. cancel-sign-sub-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. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. distribute-rgt-in N/A

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

   6. associate-+r+ N/A

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

   7. associate-*r* N/A

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

   8. metadata-eval N/A

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

   9. associate-+r+ N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-in N/A

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

   12. +-commutative N/A

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

   13. +-commutative N/A

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

5. Simplified 100.0%

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

Alternative 2: 11.1% accurate, 11.6× speedup

Math

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

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(y * Float64(y * 2.0))
end

MATLAB

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

Wolfram

code[x_, y_] := N[(y * N[(y * 2.0), $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. 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. distribute-rgt-neg-in N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. neg-sub0 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. +-commutative N/A

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

   13. associate--r+ N/A

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

   14. metadata-eval N/A

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

   15. lower--.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 1.5

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

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

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

   1. metadata-eval N/A

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

   2. lft-mult-inverse N/A

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

   3. associate-*l* N/A

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

   4. associate-*r* N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. unpow2 N/A

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

   8. unpow3 N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. associate-*l* N/A

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

   13. unpow3 N/A

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

   14. unpow2 N/A

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

   15. associate-*r* N/A

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

   16. lft-mult-inverse N/A

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

   17. *-lft-identity N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 11.1

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

8. Simplified 11.1%

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

Reproduce

herbie shell --seed 2024215 
(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))))