ENA, Section 1.4, Mentioned, B

Percentage Accurate: 87.8% → 99.6%

Time: 12.9s

Alternatives: 4

Speedup: 1.1×

Specification

\[0.999 \leq x \land x \leq 1.001\]

Math

\[\begin{array}{l} \\ \frac{10}{1 - x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 10.0 (- 1.0 (* x x))))

C

double code(double x) {
	return 10.0 / (1.0 - (x * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 10.0d0 / (1.0d0 - (x * x))
end function

Java

public static double code(double x) {
	return 10.0 / (1.0 - (x * x));
}

Python

def code(x):
	return 10.0 / (1.0 - (x * x))

Julia

function code(x)
	return Float64(10.0 / Float64(1.0 - Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = 10.0 / (1.0 - (x * x));
end

Wolfram

code[x_] := N[(10.0 / N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 87.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{10}{1 - x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 10.0 (- 1.0 (* x x))))

C

double code(double x) {
	return 10.0 / (1.0 - (x * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 10.0d0 / (1.0d0 - (x * x))
end function

Java

public static double code(double x) {
	return 10.0 / (1.0 - (x * x));
}

Python

def code(x):
	return 10.0 / (1.0 - (x * x))

Julia

function code(x)
	return Float64(10.0 / Float64(1.0 - Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = 10.0 / (1.0 - (x * x));
end

Wolfram

code[x_] := N[(10.0 / N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{-10}{\mathsf{fma}\left(x, x, -1\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -10.0 (fma x x -1.0)))

C

double code(double x) {
	return -10.0 / fma(x, x, -1.0);
}

Julia

function code(x)
	return Float64(-10.0 / fma(x, x, -1.0))
end

Wolfram

code[x_] := N[(-10.0 / N[(x * x + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.4%

   \[\frac{10}{1 - x \cdot x} \]
2. Add Preprocessing

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{-10}{\mathsf{fma}\left(x, x, -1\right)}} \]
5. Add Preprocessing

Alternative 2: 18.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{10}{1 - x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 10.0 (- 1.0 x)))

C

double code(double x) {
	return 10.0 / (1.0 - x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 10.0d0 / (1.0d0 - x)
end function

Java

public static double code(double x) {
	return 10.0 / (1.0 - x);
}

Python

def code(x):
	return 10.0 / (1.0 - x)

Julia

function code(x)
	return Float64(10.0 / Float64(1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = 10.0 / (1.0 - x);
end

Wolfram

code[x_] := N[(10.0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.8%

   \[\frac{10}{1 - x \cdot x} \]
2. Add Preprocessing

4. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{-10}{\mathsf{fma}\left(x, x, -1\right)}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-10}{\mathsf{fma}\left(x, x, -1\right)}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(10\right)}}{\mathsf{fma}\left(x, x, -1\right)} \]

   3. distribute-neg-frac N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{10}{\mathsf{fma}\left(x, x, -1\right)}\right)} \]

   4. lift-fma.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{10}{\color{blue}{x \cdot x + -1}}\right) \]

   5. difference-of-sqr--1 N/A

      \[\leadsto \mathsf{neg}\left(\frac{10}{\color{blue}{\left(x + 1\right) \cdot \left(x - 1\right)}}\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{\frac{10}{x + 1}}{x - 1}}\right) \]

   7. distribute-neg-frac2 N/A

      \[\leadsto \color{blue}{\frac{\frac{10}{x + 1}}{\mathsf{neg}\left(\left(x - 1\right)\right)}} \]

   8. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{10}{x + 1}}{\mathsf{neg}\left(\left(x - 1\right)\right)}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{10}{x + 1}}}{\mathsf{neg}\left(\left(x - 1\right)\right)} \]

   10. lower-+.f64 N/A

       \[\leadsto \frac{\frac{10}{\color{blue}{x + 1}}}{\mathsf{neg}\left(\left(x - 1\right)\right)} \]

   11. flip-- N/A

       \[\leadsto \frac{\frac{10}{x + 1}}{\mathsf{neg}\left(\color{blue}{\frac{x \cdot x - 1 \cdot 1}{x + 1}}\right)} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\frac{10}{x + 1}}{\mathsf{neg}\left(\frac{x \cdot x - \color{blue}{1}}{x + 1}\right)} \]

   13. difference-of-sqr-1 N/A

       \[\leadsto \frac{\frac{10}{x + 1}}{\mathsf{neg}\left(\frac{\color{blue}{\left(x + 1\right) \cdot \left(x - 1\right)}}{x + 1}\right)} \]

   14. difference-of-sqr--1 N/A

       \[\leadsto \frac{\frac{10}{x + 1}}{\mathsf{neg}\left(\frac{\color{blue}{x \cdot x + -1}}{x + 1}\right)} \]

   15. lift-fma.f64 N/A

       \[\leadsto \frac{\frac{10}{x + 1}}{\mathsf{neg}\left(\frac{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}{x + 1}\right)} \]

   16. distribute-neg-frac N/A

       \[\leadsto \frac{\frac{10}{x + 1}}{\color{blue}{\frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x, -1\right)\right)}{x + 1}}} \]

   17. lift-fma.f64 N/A

       \[\leadsto \frac{\frac{10}{x + 1}}{\frac{\mathsf{neg}\left(\color{blue}{\left(x \cdot x + -1\right)}\right)}{x + 1}} \]

   18. lift-*.f64 N/A

       \[\leadsto \frac{\frac{10}{x + 1}}{\frac{\mathsf{neg}\left(\left(\color{blue}{x \cdot x} + -1\right)\right)}{x + 1}} \]

   19. +-commutative N/A

       \[\leadsto \frac{\frac{10}{x + 1}}{\frac{\mathsf{neg}\left(\color{blue}{\left(-1 + x \cdot x\right)}\right)}{x + 1}} \]

   20. distribute-neg-in N/A

       \[\leadsto \frac{\frac{10}{x + 1}}{\frac{\color{blue}{\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(x \cdot x\right)\right)}}{x + 1}} \]

   21. metadata-eval N/A

       \[\leadsto \frac{\frac{10}{x + 1}}{\frac{\color{blue}{1} + \left(\mathsf{neg}\left(x \cdot x\right)\right)}{x + 1}} \]

   22. sub-neg N/A

       \[\leadsto \frac{\frac{10}{x + 1}}{\frac{\color{blue}{1 - x \cdot x}}{x + 1}} \]

   23. lift-*.f64 N/A

       \[\leadsto \frac{\frac{10}{x + 1}}{\frac{1 - \color{blue}{x \cdot x}}{x + 1}} \]

   24. metadata-eval N/A

       \[\leadsto \frac{\frac{10}{x + 1}}{\frac{\color{blue}{1 \cdot 1} - x \cdot x}{x + 1}} \]

   25. +-commutative N/A

       \[\leadsto \frac{\frac{10}{x + 1}}{\frac{1 \cdot 1 - x \cdot x}{\color{blue}{1 + x}}} \]

6. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\frac{10}{x + 1}}{1 - x}} \]

7. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{10}}{1 - x} \]
8. Step-by-step derivation

9. Applied rewrites 18.8%

   \[\leadsto \frac{\color{blue}{10}}{1 - x} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024223 
(FPCore (x)
  :name "ENA, Section 1.4, Mentioned, B"
  :precision binary64
  :pre (and (<= 0.999 x) (<= x 1.001))
  (/ 10.0 (- 1.0 (* x x))))