ENA, Section 1.4, Mentioned, B

Percentage Accurate: 87.7% → 99.6%

Time: 8.7s

Alternatives: 6

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 87.7%

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

   1. sqr-neg 87.7%

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

   2. sub-neg 87.7%

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

   3. +-commutative 87.7%

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

   4. neg-sub0 87.7%

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

   5. associate-+l- 87.7%

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

   6. sub0-neg 87.7%

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

   7. neg-mul-1 87.7%

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

   8. associate-/r* 87.7%

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

   9. metadata-eval 87.7%

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

   10. sqr-neg 87.7%

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

   11. fma-neg 99.7%

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

   12. metadata-eval 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.7%

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

   1. sub-neg 87.7%

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

   2. metadata-eval 87.7%

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

   3. distribute-neg-in 87.7%

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

   4. +-commutative 87.7%

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

   5. difference-of-sqr--1 99.4%

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

   6. distribute-rgt-neg-in 99.4%

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

   7. sub-neg 99.4%

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

   8. metadata-eval 99.4%

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

3. Applied egg-rr 99.4%

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

   1. neg-sub0 99.4%

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

   2. +-commutative 99.4%

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

   3. associate--r+ 99.4%

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

   4. metadata-eval 99.4%

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

5. Simplified 99.4%

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

   1. *-commutative 99.4%

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

   2. +-commutative 99.4%

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

   3. distribute-lft-in 99.6%

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

   4. *-commutative 99.6%

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

   5. *-un-lft-identity 99.6%

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

7. Applied egg-rr 99.6%

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

8. Final simplification 99.6%

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

Alternative 3: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.7%

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

   1. sub-neg 87.7%

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

   2. metadata-eval 87.7%

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

   3. distribute-neg-in 87.7%

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

   4. +-commutative 87.7%

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

   5. difference-of-sqr--1 99.4%

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

   6. distribute-rgt-neg-in 99.4%

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

   7. sub-neg 99.4%

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

   8. metadata-eval 99.4%

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

3. Applied egg-rr 99.4%

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

   1. neg-sub0 99.4%

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

   2. +-commutative 99.4%

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

   3. associate--r+ 99.4%

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

   4. metadata-eval 99.4%

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

5. Simplified 99.4%

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

6. Final simplification 99.4%

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

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 87.7%

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

2. Final simplification 87.7%

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

Alternative 5: 11.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.7%

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

   1. sub-neg 87.7%

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

   2. metadata-eval 87.7%

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

   3. distribute-neg-in 87.7%

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

   4. +-commutative 87.7%

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

   5. difference-of-sqr--1 99.4%

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

   6. distribute-rgt-neg-in 99.4%

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

   7. sub-neg 99.4%

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

   8. metadata-eval 99.4%

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

3. Applied egg-rr 99.4%

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

   1. neg-sub0 99.4%

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

   2. +-commutative 99.4%

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

   3. associate--r+ 99.4%

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

   4. metadata-eval 99.4%

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

5. Simplified 99.4%

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

   1. associate-/l/ 99.4%

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

   2. flip-+ 87.7%

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

   3. metadata-eval 87.7%

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

   4. fma-neg 99.3%

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

   5. metadata-eval 99.3%

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

   6. associate-/r/ 99.4%

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

   7. sub-neg 99.4%

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

   8. metadata-eval 99.4%

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

   9. +-commutative 99.4%

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

7. Applied egg-rr 99.4%

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

8. Taylor expanded in x around 0 11.8%

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

9. Final simplification 11.8%

   \[\leadsto -10 \cdot \left(x + -1\right) \]

Alternative 6: 9.5% accurate, 7.0× 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 87.7%

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

   1. sqr-neg 87.7%

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

   2. sub-neg 87.7%

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

   3. +-commutative 87.7%

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

   4. neg-sub0 87.7%

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

   5. associate-+l- 87.7%

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

   6. sub0-neg 87.7%

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

   7. neg-mul-1 87.7%

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

   8. associate-/r* 87.7%

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

   9. metadata-eval 87.7%

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

   10. sqr-neg 87.7%

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

   11. fma-neg 99.7%

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

   12. metadata-eval 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in x around 0 9.7%

   \[\leadsto \color{blue}{10} \]

5. Final simplification 9.7%

   \[\leadsto 10 \]

Reproduce

herbie shell --seed 2023301 
(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))))