From Rump in a 1983 paper

Percentage Accurate: 18.8% → 100.0%

Time: 3.7s

Alternatives: 2

Speedup: 9.3×

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, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(x \cdot x\right)\\ \left(t\_0 + y \cdot y\right) \cdot \left(t\_0 - y \cdot y\right) + \left(y \cdot y\right) \cdot 2 \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 3.0 (* x x))))
   (+ (* (+ t_0 (* y y)) (- t_0 (* y y))) (* (* y y) 2.0))))

C

double code(double x, double y) {
	double t_0 = 3.0 * (x * x);
	return ((t_0 + (y * y)) * (t_0 - (y * y))) + ((y * y) * 2.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = 3.0d0 * (x * x)
    code = ((t_0 + (y * y)) * (t_0 - (y * y))) + ((y * y) * 2.0d0)
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 * (x * x);
	return ((t_0 + (y * y)) * (t_0 - (y * y))) + ((y * y) * 2.0);
}

Python

def code(x, y):
	t_0 = 3.0 * (x * x)
	return ((t_0 + (y * y)) * (t_0 - (y * y))) + ((y * y) * 2.0)

Julia

function code(x, y)
	t_0 = Float64(3.0 * Float64(x * x))
	return Float64(Float64(Float64(t_0 + Float64(y * y)) * Float64(t_0 - Float64(y * y))) + Float64(Float64(y * y) * 2.0))
end

MATLAB

function tmp = code(x, y)
	t_0 = 3.0 * (x * x);
	tmp = ((t_0 + (y * y)) * (t_0 - (y * y))) + ((y * y) * 2.0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(t$95$0 + N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] * 2.0), $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. add-sqr-sqrt 18.8%

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

   2. sqr-pow 18.8%

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

   3. difference-of-squares 100.0%

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

   4. sqrt-prod 100.0%

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

   5. metadata-eval 100.0%

      \[\leadsto \left(\color{blue}{3} \cdot \sqrt{{x}^{4}} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left(\sqrt{9 \cdot {x}^{4}} - {y}^{\left(\frac{4}{2}\right)}\right) + 2 \cdot \left(y \cdot y\right) \]

   6. sqrt-pow1 100.0%

      \[\leadsto \left(3 \cdot \color{blue}{{x}^{\left(\frac{4}{2}\right)}} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left(\sqrt{9 \cdot {x}^{4}} - {y}^{\left(\frac{4}{2}\right)}\right) + 2 \cdot \left(y \cdot y\right) \]

   7. metadata-eval 100.0%

      \[\leadsto \left(3 \cdot {x}^{\color{blue}{2}} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left(\sqrt{9 \cdot {x}^{4}} - {y}^{\left(\frac{4}{2}\right)}\right) + 2 \cdot \left(y \cdot y\right) \]

   8. pow2 100.0%

      \[\leadsto \left(3 \cdot \color{blue}{\left(x \cdot x\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left(\sqrt{9 \cdot {x}^{4}} - {y}^{\left(\frac{4}{2}\right)}\right) + 2 \cdot \left(y \cdot y\right) \]

   9. metadata-eval 100.0%

      \[\leadsto \left(3 \cdot \left(x \cdot x\right) + {y}^{\color{blue}{2}}\right) \cdot \left(\sqrt{9 \cdot {x}^{4}} - {y}^{\left(\frac{4}{2}\right)}\right) + 2 \cdot \left(y \cdot y\right) \]

   10. pow2 100.0%

       \[\leadsto \left(3 \cdot \left(x \cdot x\right) + \color{blue}{y \cdot y}\right) \cdot \left(\sqrt{9 \cdot {x}^{4}} - {y}^{\left(\frac{4}{2}\right)}\right) + 2 \cdot \left(y \cdot y\right) \]

   11. sqrt-prod 100.0%

       \[\leadsto \left(3 \cdot \left(x \cdot x\right) + y \cdot y\right) \cdot \left(\color{blue}{\sqrt{9} \cdot \sqrt{{x}^{4}}} - {y}^{\left(\frac{4}{2}\right)}\right) + 2 \cdot \left(y \cdot y\right) \]

   12. metadata-eval 100.0%

       \[\leadsto \left(3 \cdot \left(x \cdot x\right) + y \cdot y\right) \cdot \left(\color{blue}{3} \cdot \sqrt{{x}^{4}} - {y}^{\left(\frac{4}{2}\right)}\right) + 2 \cdot \left(y \cdot y\right) \]

   13. sqrt-pow1 100.0%

       \[\leadsto \left(3 \cdot \left(x \cdot x\right) + y \cdot y\right) \cdot \left(3 \cdot \color{blue}{{x}^{\left(\frac{4}{2}\right)}} - {y}^{\left(\frac{4}{2}\right)}\right) + 2 \cdot \left(y \cdot y\right) \]

   14. metadata-eval 100.0%

       \[\leadsto \left(3 \cdot \left(x \cdot x\right) + y \cdot y\right) \cdot \left(3 \cdot {x}^{\color{blue}{2}} - {y}^{\left(\frac{4}{2}\right)}\right) + 2 \cdot \left(y \cdot y\right) \]

   15. pow2 100.0%

       \[\leadsto \left(3 \cdot \left(x \cdot x\right) + y \cdot y\right) \cdot \left(3 \cdot \color{blue}{\left(x \cdot x\right)} - {y}^{\left(\frac{4}{2}\right)}\right) + 2 \cdot \left(y \cdot y\right) \]

   16. metadata-eval 100.0%

       \[\leadsto \left(3 \cdot \left(x \cdot x\right) + y \cdot y\right) \cdot \left(3 \cdot \left(x \cdot x\right) - {y}^{\color{blue}{2}}\right) + 2 \cdot \left(y \cdot y\right) \]

   17. pow2 100.0%

       \[\leadsto \left(3 \cdot \left(x \cdot x\right) + y \cdot y\right) \cdot \left(3 \cdot \left(x \cdot x\right) - \color{blue}{y \cdot y}\right) + 2 \cdot \left(y \cdot y\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(3 \cdot \left(x \cdot x\right) + y \cdot y\right) \cdot \left(3 \cdot \left(x \cdot x\right) - y \cdot y\right)} + 2 \cdot \left(y \cdot y\right) \]

5. Final simplification 100.0%

   \[\leadsto \left(3 \cdot \left(x \cdot x\right) + y \cdot y\right) \cdot \left(3 \cdot \left(x \cdot x\right) - y \cdot y\right) + \left(y \cdot y\right) \cdot 2 \]
6. Add Preprocessing

Alternative 2: 18.8% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(N[(y * y), $MachinePrecision] * 2.0), $MachinePrecision] + N[(N[(9.0 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(y * y), $MachinePrecision] * N[(y * y), $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. metadata-eval 18.8%

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

   2. pow-sqr 18.8%

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

   3. pow2 18.8%

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

   4. pow2 18.8%

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

   5. metadata-eval 18.8%

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

   6. pow-sqr 18.8%

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

   7. pow2 18.8%

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

   8. pow2 18.8%

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

4. Applied egg-rr 18.8%

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

5. Final simplification 18.8%

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

Reproduce

herbie shell --seed 2024097 
(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))))