ENA, Section 1.4, Mentioned, B

Percentage Accurate: 87.7% → 99.6%

Time: 12.5s

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 87.8%

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

4. Applied rewrites 99.7%

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

Alternative 2: 13.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{10}{1 - x \cdot x} \leq -5000:\\ \;\;\;\;-10\\ \mathbf{else}:\\ \;\;\;\;10\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ 10.0 (- 1.0 (* x x))) -5000.0) -10.0 10.0))

C

double code(double x) {
	double tmp;
	if ((10.0 / (1.0 - (x * x))) <= -5000.0) {
		tmp = -10.0;
	} else {
		tmp = 10.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((10.0d0 / (1.0d0 - (x * x))) <= (-5000.0d0)) then
        tmp = -10.0d0
    else
        tmp = 10.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((10.0 / (1.0 - (x * x))) <= -5000.0) {
		tmp = -10.0;
	} else {
		tmp = 10.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (10.0 / (1.0 - (x * x))) <= -5000.0:
		tmp = -10.0
	else:
		tmp = 10.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(10.0 / Float64(1.0 - Float64(x * x))) <= -5000.0)
		tmp = -10.0;
	else
		tmp = 10.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((10.0 / (1.0 - (x * x))) <= -5000.0)
		tmp = -10.0;
	else
		tmp = 10.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(10.0 / N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5000.0], -10.0, 10.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 10 binary64) (-.f64 #s(literal 1 binary64) (*.f64 x x))) < -5e3

   1. Initial program 87.0%

      \[\frac{10}{1 - x \cdot x} \]
   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 1.5

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

   5. Applied rewrites 1.5%

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

   7. Applied rewrites 1.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto -10 \]
   9. Step-by-step derivation

   10. Applied rewrites 13.5%

       \[\leadsto -10 \]

   if -5e3 < (/.f64 #s(literal 10 binary64) (-.f64 #s(literal 1 binary64) (*.f64 x x)))

   1. Initial program 88.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 13.5%

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

Alternative 3: 13.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.2%

      \[\frac{10}{1 - x \cdot x} \]
   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 13.7

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

   5. Applied rewrites 13.7%

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

   7. Applied rewrites 13.7%

      \[\leadsto 10 \cdot \left(x \cdot x\right) + \color{blue}{10} \]
   8. Step-by-step derivation

   9. Applied rewrites 13.7%

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

   if 1 < (*.f64 x x)

   1. Initial program 87.0%

      \[\frac{10}{1 - x \cdot x} \]
   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 1.5

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

   5. Applied rewrites 1.5%

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

   7. Applied rewrites 1.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto -10 \]
   9. Step-by-step derivation

   10. Applied rewrites 13.5%

       \[\leadsto -10 \]
3. Recombined 2 regimes into one program.

4. Final simplification 13.6%

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

Alternative 4: 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 99.7%

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

   1. lift-fma.f64 N/A

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

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

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

   3. *-commutative N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. /-rgt-identity N/A

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

   9. lower-+.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-+.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

   16. lower-+.f64 99.5

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

6. Applied rewrites 99.5%

   \[\leadsto \frac{-10}{\color{blue}{\left(x + -1\right) + \left(x + -1\right) \cdot x}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. distribute-rgt1-in N/A

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

   6. lift-+.f64 N/A

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

   7. associate-/r* N/A

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

   8. frac-2neg N/A

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

   9. div-inv N/A

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

   10. distribute-lft-neg-in N/A

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

   11. metadata-eval N/A

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

   12. un-div-inv N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. distribute-neg-in N/A

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

   18. metadata-eval N/A

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

   19. sub-neg N/A

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

   20. lower--.f64 99.4

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

8. Applied rewrites 99.4%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 18.8%

    \[\leadsto \frac{\color{blue}{10}}{1 - x} \]
12. 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 87.8%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 9.4%

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

Reproduce

herbie shell --seed 2024219 
(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))))