From Rump in a 1983 paper

Percentage Accurate: 18.8% → 100.0%

Time: 6.0s

Alternatives: 5

Speedup: 1.6×

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, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	return Float64(Float64(2.0 * Float64(y * y)) + fma(Float64(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[(N[((-y) * y), $MachinePrecision] * N[(y * y), $MachinePrecision] + N[(9.0 * N[Power[x, 4.0], $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

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

   2. lift-pow.f64 N/A

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

   3. sqr-pow N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 18.8

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

4. Applied rewrites 18.8%

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

6. Applied rewrites 21.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \frac{{x}^{9} \cdot 729}{\mathsf{fma}\left({\left(x \cdot y\right)}^{4}, 9, \mathsf{fma}\left({x}^{8}, 81, {y}^{8}\right)\right)}, -\frac{{y}^{12}}{\mathsf{fma}\left({\left(x \cdot y\right)}^{4}, 9, \mathsf{fma}\left({x}^{8}, 81, {y}^{8}\right)\right)}\right)} + 2 \cdot \left(y \cdot y\right) \]

8. Applied rewrites 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 18.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. lift-pow.f64 N/A

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

   3. sqr-pow N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 18.8

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

4. Applied rewrites 18.8%

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

Alternative 3: 11.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift-pow.f64 N/A

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

   3. sqr-pow N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 18.8

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

4. Applied rewrites 18.8%

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

6. Applied rewrites 11.1%

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

Alternative 4: 9.6% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return (((x * x) * 9.0) * (x * x)) - (2.0 * (y * y))

Julia

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

MATLAB

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

Wolfram

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

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-pow.f64 1.5

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

5. Applied rewrites 1.5%

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

6. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 9.6

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

8. Applied rewrites 9.6%

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

10. Applied rewrites 9.6%

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

    1. lift-*.f64 N/A

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

    2. fabs-sqr-rev N/A

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

    3. lift-*.f64 N/A

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

    4. neg-fabs N/A

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

    5. lift-*.f64 N/A

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

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

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

    7. lift-neg.f64 N/A

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

    8. lift-*.f64 N/A

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

    9. unpow1 N/A

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

    10. sqr-pow N/A

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

    11. fabs-sqr-rev N/A

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

    12. sqr-pow N/A

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

    13. unpow1 9.6

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

    14. lift-*.f64 N/A

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

    15. lift-neg.f64 N/A

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

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

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

    17. lift-*.f64 N/A

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

    18. lower-neg.f64 9.6

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

12. Applied rewrites 9.6%

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

13. Final simplification 9.6%

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

Alternative 5: 9.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot x\right) \cdot 9\right) \cdot \left(x \cdot x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (* (* x x) 9.0) (* x x)))

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return ((x * x) * 9.0) * (x * x);
}

Python

def code(x, y):
	return ((x * x) * 9.0) * (x * x)

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = ((x * x) * 9.0) * (x * x);
end

Wolfram

code[x_, y_] := N[(N[(N[(x * x), $MachinePrecision] * 9.0), $MachinePrecision] * N[(x * x), $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

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-pow.f64 1.5

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

5. Applied rewrites 1.5%

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

6. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 9.6

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

8. Applied rewrites 9.6%

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

10. Applied rewrites 9.6%

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

    1. lift-*.f64 N/A

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

    2. count-2-rev N/A

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

    3. lift-*.f64 N/A

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

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

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

    5. lift-neg.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. unpow1 N/A

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

    8. sqr-pow N/A

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

    9. fabs-sqr-rev N/A

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

    10. sqr-pow N/A

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

    11. unpow1 N/A

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

    12. lift-*.f64 N/A

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

    13. lift-neg.f64 N/A

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

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

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

    15. lift-*.f64 N/A

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

    16. neg-fabs N/A

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

    17. lift-*.f64 N/A

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

    18. fabs-sqr-rev N/A

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

    19. lift-*.f64 N/A

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

    20. +-inverses 9.6

        \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 9\right) + \color{blue}{0} \]

12. Applied rewrites 9.6%

    \[\leadsto \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 9\right) + \color{blue}{0} \]

13. Final simplification 9.6%

    \[\leadsto \left(\left(x \cdot x\right) \cdot 9\right) \cdot \left(x \cdot x\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024312 
(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))))