ENA, Section 1.4, Mentioned, B

Percentage Accurate: 87.7% → 99.6%

Time: 7.9s

Alternatives: 4

Speedup: 1.0×

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, 0.1× 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 87.3%

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

   1. sqr-neg 87.3%

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

   2. remove-double-neg 87.3%

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

   3. distribute-neg-frac2 87.3%

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

   4. distribute-neg-frac 87.3%

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

   5. metadata-eval 87.3%

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

   6. neg-sub0 87.3%

      \[\leadsto \frac{-10}{\color{blue}{0 - \left(1 - \left(-x\right) \cdot \left(-x\right)\right)}} \]

   7. associate--r- 87.3%

      \[\leadsto \frac{-10}{\color{blue}{\left(0 - 1\right) + \left(-x\right) \cdot \left(-x\right)}} \]

   8. metadata-eval 87.3%

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

   9. +-commutative 87.3%

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

   10. sqr-neg 87.3%

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

   11. fma-define 99.6%

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

3. Simplified 99.6%

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

Alternative 2: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

   1. sqr-neg 87.3%

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

   2. remove-double-neg 87.3%

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

   3. distribute-neg-frac2 87.3%

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

   4. distribute-neg-frac 87.3%

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

   5. metadata-eval 87.3%

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

   6. neg-sub0 87.3%

      \[\leadsto \frac{-10}{\color{blue}{0 - \left(1 - \left(-x\right) \cdot \left(-x\right)\right)}} \]

   7. associate--r- 87.3%

      \[\leadsto \frac{-10}{\color{blue}{\left(0 - 1\right) + \left(-x\right) \cdot \left(-x\right)}} \]

   8. metadata-eval 87.3%

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

   9. +-commutative 87.3%

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

   10. sqr-neg 87.3%

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

   11. fma-define 99.6%

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

3. Simplified 99.6%

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

   1. frac-2neg 99.6%

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

   2. metadata-eval 99.6%

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

   3. div-inv 99.5%

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

   4. fma-undefine 87.3%

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

   5. distribute-neg-in 87.3%

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

   6. metadata-eval 87.3%

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

   7. +-commutative 87.3%

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

   8. sub-neg 87.3%

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

   9. *-commutative 87.3%

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

6. Applied egg-rr 99.5%

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

   1. associate-*l/ 99.6%

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

   2. metadata-eval 99.6%

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

   3. fma-undefine 87.3%

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

   4. difference-of-sqr--1 99.5%

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

   5. *-commutative 99.5%

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

   6. associate-/r* 99.5%

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

   7. sub-neg 99.5%

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

   8. metadata-eval 99.5%

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

   9. +-commutative 99.5%

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

8. Applied egg-rr 99.5%

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

9. Final simplification 99.5%

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

Alternative 3: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{-10}{\left(x + 1\right) \cdot \left(x + -1\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

   1. sqr-neg 87.3%

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

   2. remove-double-neg 87.3%

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

   3. distribute-neg-frac2 87.3%

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

   4. distribute-neg-frac 87.3%

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

   5. metadata-eval 87.3%

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

   6. neg-sub0 87.3%

      \[\leadsto \frac{-10}{\color{blue}{0 - \left(1 - \left(-x\right) \cdot \left(-x\right)\right)}} \]

   7. associate--r- 87.3%

      \[\leadsto \frac{-10}{\color{blue}{\left(0 - 1\right) + \left(-x\right) \cdot \left(-x\right)}} \]

   8. metadata-eval 87.3%

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

   9. +-commutative 87.3%

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

   10. sqr-neg 87.3%

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

   11. fma-define 99.6%

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

3. Simplified 99.6%

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

   1. fma-undefine 87.3%

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

   2. difference-of-sqr--1 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

   \[\leadsto \frac{-10}{\left(x + 1\right) \cdot \left(x + -1\right)} \]
8. Add Preprocessing

Alternative 4: 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]

Derivation

1. Initial program 87.3%

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

Reproduce

herbie shell --seed 2024179 
(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))))