ENA, Section 1.4, Mentioned, B

Percentage Accurate: 87.8% → 99.6%

Time: 10.1s

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 87.5%

   \[\frac{10}{1 - x \cdot x} \]
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: 13.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - x \cdot x \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{-10}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 10, 10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- 1.0 (* x x)) -2e-5) (/ -10.0 (* x x)) (fma (* x x) 10.0 10.0)))

C

double code(double x) {
	double tmp;
	if ((1.0 - (x * x)) <= -2e-5) {
		tmp = -10.0 / (x * x);
	} else {
		tmp = fma((x * x), 10.0, 10.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 - Float64(x * x)) <= -2e-5)
		tmp = Float64(-10.0 / Float64(x * x));
	else
		tmp = fma(Float64(x * x), 10.0, 10.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision], -2e-5], N[(-10.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * 10.0 + 10.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 x x)) < -2.00000000000000016e-5

   1. Initial program 86.2%

      \[\frac{10}{1 - x \cdot x} \]
   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%

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

   if -2.00000000000000016e-5 < (-.f64 #s(literal 1 binary64) (*.f64 x x))

   1. Initial program 88.0%

      \[\frac{10}{1 - x \cdot x} \]
   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. *-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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, 10, 10\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 9.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.5%

   \[\frac{10}{1 - x \cdot x} \]
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 9.8

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

5. Applied rewrites 9.8%

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

Alternative 4: 9.4% 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 87.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 9.6%

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

Reproduce

herbie shell --seed 2024240 
(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))))