Examples.Basics.BasicTests:f2 from sbv-4.4

Percentage Accurate: 93.8% → 100.0%

Time: 5.2s

Alternatives: 1

Speedup: 1.0×

• 43.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot x - y \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision]

Initial Program: 93.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot x - y \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(x + y), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.3%

   \[x \cdot x - y \cdot y \]
2. Add Preprocessing
3. Step-by-step derivation

   1. difference-of-squares 100.0%

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

   2. add-sqr-sqrt 49.5%

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

   3. sqrt-prod 79.0%

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

   4. sqr-neg 79.0%

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

   5. sqrt-unprod 29.8%

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

   6. add-sqr-sqrt 49.6%

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

   7. sub-neg 49.6%

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

   8. pow1 49.6%

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

   9. pow1 49.6%

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

   10. pow-prod-up 49.6%

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

   11. add-sqr-sqrt 26.4%

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

   12. add-sqr-sqrt 10.2%

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

   13. difference-of-squares 10.2%

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

   14. metadata-eval 10.2%

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

   15. unpow-prod-down 10.2%

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

4. Applied egg-rr 10.2%

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

   1. unpow2 10.2%

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

   2. unpow2 10.2%

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

   3. unswap-sqr 10.2%

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

   4. difference-of-squares 10.2%

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

   5. unpow1/2 10.2%

      \[\leadsto \left(\color{blue}{{x}^{0.5}} \cdot \sqrt{x} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

   6. unpow1/2 10.2%

      \[\leadsto \left({x}^{0.5} \cdot \color{blue}{{x}^{0.5}} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

   7. pow-sqr 10.2%

      \[\leadsto \left(\color{blue}{{x}^{\left(2 \cdot 0.5\right)}} - \sqrt{y} \cdot \sqrt{y}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

   8. metadata-eval 10.2%

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

   9. unpow1 10.2%

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

   10. unpow1/2 10.2%

       \[\leadsto \left(x - \color{blue}{{y}^{0.5}} \cdot \sqrt{y}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

   11. unpow1/2 10.2%

       \[\leadsto \left(x - {y}^{0.5} \cdot \color{blue}{{y}^{0.5}}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

   12. pow-sqr 10.2%

       \[\leadsto \left(x - \color{blue}{{y}^{\left(2 \cdot 0.5\right)}}\right) \cdot \left(\left(\sqrt{x} + \sqrt{y}\right) \cdot \left(\sqrt{x} - \sqrt{y}\right)\right) \]

   13. metadata-eval 10.2%

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

   14. unpow1 10.2%

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

   15. difference-of-squares 10.2%

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

   16. unpow1/2 10.2%

       \[\leadsto \left(x - y\right) \cdot \left(\color{blue}{{x}^{0.5}} \cdot \sqrt{x} - \sqrt{y} \cdot \sqrt{y}\right) \]

   17. unpow1/2 10.2%

       \[\leadsto \left(x - y\right) \cdot \left({x}^{0.5} \cdot \color{blue}{{x}^{0.5}} - \sqrt{y} \cdot \sqrt{y}\right) \]

   18. pow-sqr 19.8%

       \[\leadsto \left(x - y\right) \cdot \left(\color{blue}{{x}^{\left(2 \cdot 0.5\right)}} - \sqrt{y} \cdot \sqrt{y}\right) \]

   19. metadata-eval 19.8%

       \[\leadsto \left(x - y\right) \cdot \left({x}^{\color{blue}{1}} - \sqrt{y} \cdot \sqrt{y}\right) \]

   20. unpow1 19.8%

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

6. Simplified 49.6%

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

8. Applied egg-rr 22.5%

   \[\leadsto \color{blue}{\frac{{\left(x + y\right)}^{2} \cdot \left({x}^{3} - {y}^{3}\right)}{\left(x + y\right) \cdot \mathsf{fma}\left(y, x + y, {x}^{2}\right)}} \]
9. Step-by-step derivation

   1. times-frac 37.0%

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

   2. associate-*r/ 30.0%

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

   3. unpow2 30.0%

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

   4. associate-/l* 30.0%

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

   5. *-inverses 30.0%

      \[\leadsto \frac{\frac{x + y}{\color{blue}{1}} \cdot \left({x}^{3} - {y}^{3}\right)}{\mathsf{fma}\left(y, x + y, {x}^{2}\right)} \]

   6. /-rgt-identity 30.0%

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

   7. associate-/l* 36.9%

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

   8. associate-/r/ 36.9%

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

   9. +-commutative 36.9%

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

   10. +-commutative 36.9%

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

10. Simplified 36.9%

    \[\leadsto \color{blue}{\frac{y + x}{\mathsf{fma}\left(y, y + x, {x}^{2}\right)} \cdot \left({x}^{3} - {y}^{3}\right)} \]
11. Step-by-step derivation

    1. expm1-log1p-u 23.3%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y + x}{\mathsf{fma}\left(y, y + x, {x}^{2}\right)} \cdot \left({x}^{3} - {y}^{3}\right)\right)\right)} \]

    2. expm1-udef 12.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{y + x}{\mathsf{fma}\left(y, y + x, {x}^{2}\right)} \cdot \left({x}^{3} - {y}^{3}\right)\right)} - 1} \]

    3. associate-*l/ 9.9%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left(y + x\right) \cdot \left({x}^{3} - {y}^{3}\right)}{\mathsf{fma}\left(y, y + x, {x}^{2}\right)}}\right)} - 1 \]

    4. *-un-lft-identity 9.9%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\left(y + x\right) \cdot \left({x}^{3} - {y}^{3}\right)}{\color{blue}{1 \cdot \mathsf{fma}\left(y, y + x, {x}^{2}\right)}}\right)} - 1 \]

    5. times-frac 12.2%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{y + x}{1} \cdot \frac{{x}^{3} - {y}^{3}}{\mathsf{fma}\left(y, y + x, {x}^{2}\right)}}\right)} - 1 \]

    6. fma-udef 12.2%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{y + x}{1} \cdot \frac{{x}^{3} - {y}^{3}}{\color{blue}{y \cdot \left(y + x\right) + {x}^{2}}}\right)} - 1 \]

    7. +-commutative 12.2%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{y + x}{1} \cdot \frac{{x}^{3} - {y}^{3}}{\color{blue}{{x}^{2} + y \cdot \left(y + x\right)}}\right)} - 1 \]

    8. distribute-rgt-in 12.2%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{y + x}{1} \cdot \frac{{x}^{3} - {y}^{3}}{{x}^{2} + \color{blue}{\left(y \cdot y + x \cdot y\right)}}\right)} - 1 \]

    9. unpow2 12.2%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{y + x}{1} \cdot \frac{{x}^{3} - {y}^{3}}{\color{blue}{x \cdot x} + \left(y \cdot y + x \cdot y\right)}\right)} - 1 \]

    10. flip3-- 40.4%

        \[\leadsto e^{\mathsf{log1p}\left(\frac{y + x}{1} \cdot \color{blue}{\left(x - y\right)}\right)} - 1 \]

12. Applied egg-rr 40.4%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{y + x}{1} \cdot \left(x - y\right)\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 56.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y + x}{1} \cdot \left(x - y\right)\right)\right)} \]

    2. expm1-log1p 100.0%

       \[\leadsto \color{blue}{\frac{y + x}{1} \cdot \left(x - y\right)} \]

    3. /-rgt-identity 100.0%

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

    4. +-commutative 100.0%

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

14. Simplified 100.0%

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

15. Final simplification 100.0%

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

Reproduce

herbie shell --seed 2024029 
(FPCore (x y)
  :name "Examples.Basics.BasicTests:f2 from sbv-4.4"
  :precision binary64
  (- (* x x) (* y y)))