ENA, Section 1.4, Mentioned, B

Percentage Accurate: 87.8% → 99.6%

Time: 11.5s

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: 87.8% 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 88.0%

   \[\frac{10}{1 - x \cdot x} \]
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, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (*.f64 x x)) < -5.00000000000000024e-5

   1. Initial program 86.9%

      \[\frac{10}{1 - x \cdot x} \]
   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 -5.00000000000000024e-5 < (-.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

      \[\leadsto \color{blue}{10 + 10 \cdot {x}^{2}} \]
   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 13.6

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

   5. Applied rewrites 13.6%

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

   7. Applied rewrites 13.6%

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

   9. Applied rewrites 13.6%

      \[\leadsto \mathsf{fma}\left(x, x, 1\right) \cdot \color{blue}{10} \]
3. Recombined 2 regimes into one program.

4. Final simplification 13.6%

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

Reproduce

herbie shell --seed 2024228 
(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))))