From Rump in a 1983 paper

Average Error: 52.0 → 0

Time: 2.7s

Precision: binary64

Cost: 13888

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

Math

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

↓

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

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (+ (+ (+ (* 9.0 (pow x 4.0)) (- 1.0 (pow y 4.0))) -1.0) (* y (* y 2.0))))

C

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

↓

double code(double x, double y) {
	return (((9.0 * pow(x, 4.0)) + (1.0 - pow(y, 4.0))) + -1.0) + (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 ** 4.0d0)) + (2.0d0 * (y * y))
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (((9.0d0 * (x ** 4.0d0)) + (1.0d0 - (y ** 4.0d0))) + (-1.0d0)) + (y * (y * 2.0d0))
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));
}

↓

public static double code(double x, double y) {
	return (((9.0 * Math.pow(x, 4.0)) + (1.0 - Math.pow(y, 4.0))) + -1.0) + (y * (y * 2.0));
}

Python

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

↓

def code(x, y):
	return (((9.0 * math.pow(x, 4.0)) + (1.0 - math.pow(y, 4.0))) + -1.0) + (y * (y * 2.0))

Julia

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

↓

function code(x, y)
	return Float64(Float64(Float64(Float64(9.0 * (x ^ 4.0)) + Float64(1.0 - (y ^ 4.0))) + -1.0) + Float64(y * Float64(y * 2.0)))
end

MATLAB

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

↓

function tmp = code(x, y)
	tmp = (((9.0 * (x ^ 4.0)) + (1.0 - (y ^ 4.0))) + -1.0) + (y * (y * 2.0));
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]

↓

code[x_, y_] := N[(N[(N[(N[(9.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + N[(y * N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.0

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

2. Simplified 52.0

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

   [Start] 52.0	\[ \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right) \]
   rational.json-simplify-43 [=>] 52.0	\[ \left(9 \cdot {x}^{4} - {y}^{4}\right) + \color{blue}{y \cdot \left(y \cdot 2\right)} \]

3. Applied egg-rr 0

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

4. Final simplification 0

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

Alternatives

Alternative 1
Error	52.0
Cost	13632

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

Alternative 2
Error	57.8
Cost	7040

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

Alternative 3
Error	63.0
Cost	6976

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

Reproduce

herbie shell --seed 2023073 
(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))))