From Rump in a 1983 paper

Percentage Accurate: 18.8% → 100.0%

Time: 3.0s

Alternatives: 3

Speedup: 1.7×

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, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot x, 3, y \cdot y\right) \cdot \left(\left(x \cdot x\right) \cdot 3 - y \cdot y\right) + \left(y \cdot y\right) \cdot 2 \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(N[(N[(x * x), $MachinePrecision] * 3.0 + N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 3.0), $MachinePrecision] - 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. 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. metadata-eval 18.8%

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

   3. pow-prod-up 18.8%

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

   4. pow2 18.8%

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

   5. pow2 18.8%

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

   6. difference-of-squares 100.0%

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

   7. *-commutative 100.0%

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

   8. sqrt-prod 100.0%

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

   9. sqrt-pow1 100.0%

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

   10. metadata-eval 100.0%

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

   11. pow2 100.0%

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

   12. fma-def 100.0%

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

   13. metadata-eval 100.0%

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

   14. *-commutative 100.0%

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

   15. sqrt-prod 100.0%

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

   16. sqrt-pow1 100.0%

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

   17. metadata-eval 100.0%

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

   18. pow2 100.0%

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

   19. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(x \cdot x, 3, y \cdot y\right) \cdot \left(\left(x \cdot x\right) \cdot 3 - y \cdot y\right) + \left(y \cdot y\right) \cdot 2 \]

Alternative 2: 9.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

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 ** 4.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(N[(y * y), $MachinePrecision] * 2.0), $MachinePrecision] + N[(9.0 * N[Power[x, 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. Taylor expanded in x around inf 9.6%

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

3. Final simplification 9.6%

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

Alternative 3: 1.5% accurate, 23.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(y * N[(y * N[(2.0 - 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. Step-by-step derivation

   1. associate-+l- 3.1%

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

   2. fma-neg 3.1%

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

   3. sub-neg 3.1%

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

   4. +-commutative 3.1%

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

   5. distribute-neg-in 3.1%

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

   6. remove-double-neg 3.1%

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

   7. unsub-neg 3.1%

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

   8. sqr-pow 3.1%

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

   9. *-commutative 3.1%

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

   10. unpow2 3.1%

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

   11. metadata-eval 3.1%

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

   12. distribute-lft-out-- 3.1%

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

   13. metadata-eval 3.1%

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

   14. unpow2 3.1%

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

   15. metadata-eval 3.1%

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

   16. unpow2 3.1%

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

3. Simplified 3.1%

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

4. Taylor expanded in x around 0 1.5%

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

   1. unpow2 1.5%

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

   2. *-commutative 1.5%

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

   3. unpow2 1.5%

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

   4. associate-*r* 1.5%

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

6. Simplified 1.5%

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

7. Final simplification 1.5%

   \[\leadsto y \cdot \left(y \cdot \left(2 - y \cdot y\right)\right) \]

Reproduce

herbie shell --seed 2023178 
(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))))