From Rump in a 1983 paper

Percentage Accurate: 18.8% → 100.0%

Time: 4.1s

Alternatives: 5

Speedup: 1.0×

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(N[Power[y, 2.0], $MachinePrecision] * (-N[Power[y, 2.0], $MachinePrecision]) + N[(9.0 * N[Power[x, 4.0], $MachinePrecision]), $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. sub-neg 18.8%

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

   2. +-commutative 18.8%

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

   3. sqr-pow 18.8%

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

   4. distribute-rgt-neg-in 18.8%

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

   5. fma-define 100.0%

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

   6. metadata-eval 100.0%

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

   7. metadata-eval 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 18.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(y \cdot y\right) + \sqrt[3]{{\left(9 \cdot {x}^{4} - {y}^{4}\right)}^{3}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (* 2.0 (* y y)) (cbrt (pow (- (* 9.0 (pow x 4.0)) (pow y 4.0)) 3.0))))

C

double code(double x, double y) {
	return (2.0 * (y * y)) + cbrt(pow(((9.0 * pow(x, 4.0)) - pow(y, 4.0)), 3.0));
}

Java

public static double code(double x, double y) {
	return (2.0 * (y * y)) + Math.cbrt(Math.pow(((9.0 * Math.pow(x, 4.0)) - Math.pow(y, 4.0)), 3.0));
}

Julia

function code(x, y)
	return Float64(Float64(2.0 * Float64(y * y)) + cbrt((Float64(Float64(9.0 * (x ^ 4.0)) - (y ^ 4.0)) ^ 3.0)))
end

Wolfram

code[x_, y_] := N[(N[(2.0 * N[(y * y), $MachinePrecision]), $MachinePrecision] + N[Power[N[Power[N[(N[(9.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] - N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $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-cbrt-cube 18.8%

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

   2. pow3 18.8%

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

4. Applied egg-rr 18.8%

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

5. Final simplification 18.8%

   \[\leadsto 2 \cdot \left(y \cdot y\right) + \sqrt[3]{{\left(9 \cdot {x}^{4} - {y}^{4}\right)}^{3}} \]
6. Add Preprocessing

Alternative 3: 18.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(2.0 * N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(N[(9.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] - N[Power[y, 4.0], $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. Final simplification 18.8%

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

Alternative 4: 9.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(9.0 * N[Power[x, 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. Taylor expanded in x around inf 9.6%

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

4. Final simplification 9.6%

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

Alternative 5: 1.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (2.0d0 * (y * y)) - (y ** 4.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(2.0 * N[(y * y), $MachinePrecision]), $MachinePrecision] - N[Power[y, 4.0], $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. Taylor expanded in x around 0 1.5%

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

   1. neg-mul-1 1.5%

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

5. Simplified 1.5%

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

6. Final simplification 1.5%

   \[\leadsto 2 \cdot \left(y \cdot y\right) - {y}^{4} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024066 
(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))))