ENA, Section 1.4, Mentioned, B

Percentage Accurate: 87.7% → 99.6%

Time: 7.0s

Alternatives: 5

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: 87.7% 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
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. neg-mul-1 N/A

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

   5. lift-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. fabs-sqr-rev N/A

      \[\leadsto \frac{10}{1 + {-1}^{1} \cdot \color{blue}{\left|x \cdot x\right|}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{10}{1 + {-1}^{1} \cdot \left|\color{blue}{x \cdot x}\right|} \]

   9. rem-sqrt-square-rev N/A

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

   10. pow1/2 N/A

       \[\leadsto \frac{10}{1 + {-1}^{1} \cdot \color{blue}{{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}^{\frac{1}{2}}}} \]

   11. sqr-neg-rev N/A

       \[\leadsto \frac{10}{1 + {-1}^{1} \cdot {\color{blue}{\left(\left(\mathsf{neg}\left(x \cdot x\right)\right) \cdot \left(\mathsf{neg}\left(x \cdot x\right)\right)\right)}}^{\frac{1}{2}}} \]

   12. pow2 N/A

       \[\leadsto \frac{10}{1 + {-1}^{1} \cdot {\color{blue}{\left({\left(\mathsf{neg}\left(x \cdot x\right)\right)}^{2}\right)}}^{\frac{1}{2}}} \]

   13. pow-pow N/A

       \[\leadsto \frac{10}{1 + {-1}^{1} \cdot \color{blue}{{\left(\mathsf{neg}\left(x \cdot x\right)\right)}^{\left(2 \cdot \frac{1}{2}\right)}}} \]

   14. metadata-eval N/A

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

   15. unpow-prod-down N/A

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

   16. neg-mul-1 N/A

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

   17. remove-double-neg N/A

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

   18. lift-*.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. unpow1 N/A

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

   21. frac-2neg N/A

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

   22. lower-/.f64 N/A

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

4. Applied rewrites 99.6%

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

Alternative 2: 13.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x, x, 1\right) \cdot 10\\ \mathbf{else}:\\ \;\;\;\;\frac{-10}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 1.0) (* (fma x x 1.0) 10.0) (/ -10.0 (* x x))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 1.0) {
		tmp = fma(x, x, 1.0) * 10.0;
	} else {
		tmp = -10.0 / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 1.0)
		tmp = Float64(fma(x, x, 1.0) * 10.0);
	else
		tmp = Float64(-10.0 / Float64(x * x));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1

   1. Initial program 87.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{10 + 10 \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{10 \cdot {x}^{2} + 10} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot 10} + 10 \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, 10, 10\right)} \]

      4. unpow2 N/A

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

      5. lower-*.f64 13.7

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

   5. Applied rewrites 13.7%

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

   7. Applied rewrites 13.7%

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

   if 1 < (*.f64 x x)

   1. Initial program 86.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-10}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-10}{{x}^{2}}} \]

      2. unpow2 N/A

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

      3. lower-*.f64 13.5

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

   5. Applied rewrites 13.5%

      \[\leadsto \color{blue}{\frac{-10}{x \cdot x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 11.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.4%

   \[\frac{10}{1 - x \cdot x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. associate-/r/ N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(-1 \cdot {x}^{2} - 1\right)} \cdot -10 \]
6. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{\left(-1 \cdot {x}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot -10 \]

   2. metadata-eval N/A

      \[\leadsto \left(-1 \cdot {x}^{2} + \color{blue}{-1}\right) \cdot -10 \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(-1 + -1 \cdot {x}^{2}\right)} \cdot -10 \]

   4. mul-1-neg N/A

      \[\leadsto \left(-1 + \color{blue}{\left(\mathsf{neg}\left({x}^{2}\right)\right)}\right) \cdot -10 \]

   5. unsub-neg N/A

      \[\leadsto \color{blue}{\left(-1 - {x}^{2}\right)} \cdot -10 \]

   6. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(-1 - {x}^{2}\right)} \cdot -10 \]

   7. unpow2 N/A

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

   8. lower-*.f64 9.2

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

7. Applied rewrites 9.2%

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

9. Applied rewrites 11.9%

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

Alternative 4: 9.7% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* (fma x x 1.0) 10.0))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{10 + 10 \cdot {x}^{2}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{10 \cdot {x}^{2} + 10} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{{x}^{2} \cdot 10} + 10 \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, 10, 10\right)} \]

   4. unpow2 N/A

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

   5. lower-*.f64 9.2

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

5. Applied rewrites 9.2%

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

7. Applied rewrites 9.2%

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

Alternative 5: 9.5% accurate, 20.0× speedup

Math

\[\begin{array}{l} \\ 10 \end{array} \]

FPCore

(FPCore (x) :precision binary64 10.0)

C

double code(double x) {
	return 10.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 10.0

Julia

function code(x)
	return 10.0
end

MATLAB

function tmp = code(x)
	tmp = 10.0;
end

Wolfram

code[x_] := 10.0

Derivation

1. Initial program 87.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{10} \]
4. Step-by-step derivation

5. Applied rewrites 9.1%

   \[\leadsto \color{blue}{10} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024298 
(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))))