From Rump in a 1983 paper, rewritten

Average Error: 62.0 → 52.0

Time: 6.5s

Precision: binary64

Cost: 19968

\[x = 10864 \land y = 18817\]

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 * (y ** 2.0d0)) - (((x ** 4.0d0) * (-9.0d0)) + (y ** 4.0d0))
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));
}

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

↓

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

Derivation

1. Initial program 62.0

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

2. Taylor expanded in y around 0 62.0

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

3. Simplified 52.0

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

   [Start] 62.0	\[ 9 \cdot {x}^{4} + \left(2 \cdot {y}^{2} + -1 \cdot {y}^{4}\right) \]
   rational_best-simplify-47 [=>] 62.0	\[ \color{blue}{-1 \cdot {y}^{4} + \left(2 \cdot {y}^{2} + 9 \cdot {x}^{4}\right)} \]
   rational_best-simplify-3 [<=] 62.0	\[ -1 \cdot {y}^{4} + \color{blue}{\left(9 \cdot {x}^{4} + 2 \cdot {y}^{2}\right)} \]
   rational_best-simplify-3 [=>] 62.0	\[ \color{blue}{\left(9 \cdot {x}^{4} + 2 \cdot {y}^{2}\right) + -1 \cdot {y}^{4}} \]
   rational_best-simplify-59 [=>] 62.0	\[ \color{blue}{\left(2 \cdot {y}^{2} - \left(-9 \cdot {x}^{4}\right)\right)} + -1 \cdot {y}^{4} \]
   rational_best-simplify-1 [=>] 62.0	\[ \left(2 \cdot {y}^{2} - \left(-9 \cdot {x}^{4}\right)\right) + \color{blue}{{y}^{4} \cdot -1} \]
   rational_best-simplify-10 [=>] 62.0	\[ \left(2 \cdot {y}^{2} - \left(-9 \cdot {x}^{4}\right)\right) + \color{blue}{\left(-{y}^{4}\right)} \]
   rational_best-simplify-56 [=>] 52.0	\[ \color{blue}{2 \cdot {y}^{2} - \left(\left(-9 \cdot {x}^{4}\right) + {y}^{4}\right)} \]
   rational_best-simplify-11 [=>] 52.0	\[ 2 \cdot {y}^{2} - \left(\color{blue}{\left(9 \cdot {x}^{4}\right) \cdot -1} + {y}^{4}\right) \]
   rational_best-simplify-1 [=>] 52.0	\[ 2 \cdot {y}^{2} - \left(\color{blue}{-1 \cdot \left(9 \cdot {x}^{4}\right)} + {y}^{4}\right) \]
   rational_best-simplify-50 [=>] 52.0	\[ 2 \cdot {y}^{2} - \left(\color{blue}{{x}^{4} \cdot \left(9 \cdot -1\right)} + {y}^{4}\right) \]
   metadata-eval [=>] 52.0	\[ 2 \cdot {y}^{2} - \left({x}^{4} \cdot \color{blue}{-9} + {y}^{4}\right) \]

4. Final simplification 52.0

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

Alternatives

Alternative 1
Error	57.8
Cost	6656

\[9 \cdot {x}^{4} \]

Alternative 2
Error	63.0
Cost	6592

\[-{y}^{4} \]

Reproduce

herbie shell --seed 2023099 
(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))))