From Rump in a 1983 paper, rewritten

Average Error: 62.0 → 52.0

Time: 2.1s

Precision: binary64

Cost: 7552

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

Math

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

↓

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

FPCore

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

↓

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

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 (y * (y + y)) - ((y * ((y * y) * y)) + (pow(x, 4.0) * -9.0));
}

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 = (y * (y + y)) - ((y * ((y * y) * y)) + ((x ** 4.0d0) * (-9.0d0)))
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 (y * (y + y)) - ((y * ((y * y) * y)) + (Math.pow(x, 4.0) * -9.0));
}

Python

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

↓

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

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(y * Float64(y + y)) - Float64(Float64(y * Float64(Float64(y * y) * y)) + Float64((x ^ 4.0) * -9.0)))
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 = (y * (y + y)) - ((y * ((y * y) * y)) + ((x ^ 4.0) * -9.0));
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[(y * N[(y + y), $MachinePrecision]), $MachinePrecision] - N[(N[(y * N[(N[(y * y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * -9.0), $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. Applied egg-rr 52.0

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

3. Simplified 52.0

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

   [Start] 52.0	\[ y \cdot y + \left(y \cdot y - \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + {x}^{4} \cdot -9\right)\right) \]
   rational_best_oopsla_all_46_json_45_simplify-108 [=>] 52.0	\[ \color{blue}{\left(y \cdot y + y \cdot y\right) - \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + {x}^{4} \cdot -9\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-37 [<=] 52.0	\[ \color{blue}{y \cdot \left(y + y\right)} - \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + {x}^{4} \cdot -9\right) \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 52.0	\[ y \cdot \left(y + y\right) - \left(\color{blue}{y \cdot \left(\left(y \cdot y\right) \cdot y\right)} + {x}^{4} \cdot -9\right) \]

4. Final simplification 52.0

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

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 2023090 
(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))))