ENA, Section 1.4, Mentioned, B

Percentage Accurate: 87.7% → 99.6%

Time: 19.2s

Alternatives: 7

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 87.9%

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

   1. sqr-neg 87.9%

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

   2. remove-double-neg 87.9%

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

   3. distribute-neg-frac 87.9%

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

   4. distribute-frac-neg2 87.9%

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

   5. metadata-eval 87.9%

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

   6. neg-sub0 87.9%

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

   7. associate--r- 87.9%

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

   8. metadata-eval 87.9%

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

   9. +-commutative 87.9%

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

   10. sqr-neg 87.9%

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

   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: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.9%

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

   1. sqr-neg 87.9%

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

   2. remove-double-neg 87.9%

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

   3. distribute-neg-frac 87.9%

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

   4. distribute-frac-neg2 87.9%

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

   5. metadata-eval 87.9%

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

   6. neg-sub0 87.9%

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

   7. associate--r- 87.9%

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

   8. metadata-eval 87.9%

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

   9. +-commutative 87.9%

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

   10. sqr-neg 87.9%

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

   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 87.9%

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

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

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

6. Applied egg-rr 99.4%

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

   1. *-commutative 99.4%

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

   2. +-commutative 99.4%

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

   3. distribute-lft-in 99.5%

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

   4. *-commutative 99.5%

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

   5. *-un-lft-identity 99.5%

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

   6. sub-neg 99.5%

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

   7. metadata-eval 99.5%

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

   8. sub-neg 99.5%

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

   9. metadata-eval 99.5%

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

8. Applied egg-rr 99.5%

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

9. Final simplification 99.5%

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

Alternative 3: 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 87.9%

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

   1. sqr-neg 87.9%

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

   2. remove-double-neg 87.9%

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

   3. distribute-neg-frac 87.9%

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

   4. distribute-frac-neg2 87.9%

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

   5. metadata-eval 87.9%

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

   6. neg-sub0 87.9%

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

   7. associate--r- 87.9%

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

   8. metadata-eval 87.9%

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

   9. +-commutative 87.9%

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

   10. sqr-neg 87.9%

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

   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 87.9%

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

   5. distribute-neg-in 87.9%

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

   6. metadata-eval 87.9%

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

   7. +-commutative 87.9%

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

   8. sub-neg 87.9%

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

   9. *-commutative 87.9%

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

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. fma-undefine 87.9%

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

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

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

8. Applied egg-rr 99.4%

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

   1. metadata-eval 99.4%

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

   2. metadata-eval 99.4%

      \[\leadsto \frac{-1}{\left(x + 1\right) \cdot \left(x - 1\right)} \cdot {10}^{\color{blue}{\left(\frac{2}{2}\right)}} \]

   3. sqrt-pow2 98.4%

      \[\leadsto \frac{-1}{\left(x + 1\right) \cdot \left(x - 1\right)} \cdot \color{blue}{{\left(\sqrt{10}\right)}^{2}} \]

   4. associate-*l/ 98.4%

      \[\leadsto \color{blue}{\frac{-1 \cdot {\left(\sqrt{10}\right)}^{2}}{\left(x + 1\right) \cdot \left(x - 1\right)}} \]

   5. sqrt-pow2 99.4%

      \[\leadsto \frac{-1 \cdot \color{blue}{{10}^{\left(\frac{2}{2}\right)}}}{\left(x + 1\right) \cdot \left(x - 1\right)} \]

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

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

   9. *-commutative 99.4%

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

   10. associate-/r* 99.4%

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

   11. sub-neg 99.4%

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

   12. metadata-eval 99.4%

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

10. Applied egg-rr 99.4%

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

Alternative 4: 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 87.9%

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

   1. sqr-neg 87.9%

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

   2. remove-double-neg 87.9%

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

   3. distribute-neg-frac 87.9%

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

   4. distribute-frac-neg2 87.9%

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

   5. metadata-eval 87.9%

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

   6. neg-sub0 87.9%

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

   7. associate--r- 87.9%

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

   8. metadata-eval 87.9%

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

   9. +-commutative 87.9%

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

   10. sqr-neg 87.9%

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

   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 87.9%

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

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

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 5: 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 87.9%

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

Alternative 6: 18.8% accurate, 1.4× speedup

Math

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

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 87.9%

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

   1. sqr-neg 87.9%

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

   2. remove-double-neg 87.9%

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

   3. distribute-neg-frac 87.9%

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

   4. distribute-frac-neg2 87.9%

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

   5. metadata-eval 87.9%

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

   6. neg-sub0 87.9%

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

   7. associate--r- 87.9%

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

   8. metadata-eval 87.9%

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

   9. +-commutative 87.9%

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

   10. sqr-neg 87.9%

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

   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 87.9%

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

   5. distribute-neg-in 87.9%

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

   6. metadata-eval 87.9%

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

   7. +-commutative 87.9%

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

   8. sub-neg 87.9%

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

   9. *-commutative 87.9%

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

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. fma-undefine 87.9%

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

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

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

8. Applied egg-rr 99.4%

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

   1. metadata-eval 99.4%

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

   2. metadata-eval 99.4%

      \[\leadsto \frac{-1}{\left(x + 1\right) \cdot \left(x - 1\right)} \cdot {10}^{\color{blue}{\left(\frac{2}{2}\right)}} \]

   3. sqrt-pow2 98.4%

      \[\leadsto \frac{-1}{\left(x + 1\right) \cdot \left(x - 1\right)} \cdot \color{blue}{{\left(\sqrt{10}\right)}^{2}} \]

   4. associate-*l/ 98.4%

      \[\leadsto \color{blue}{\frac{-1 \cdot {\left(\sqrt{10}\right)}^{2}}{\left(x + 1\right) \cdot \left(x - 1\right)}} \]

   5. sqrt-pow2 99.4%

      \[\leadsto \frac{-1 \cdot \color{blue}{{10}^{\left(\frac{2}{2}\right)}}}{\left(x + 1\right) \cdot \left(x - 1\right)} \]

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

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

   9. associate-/r* 99.4%

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

   10. sub-neg 99.4%

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

   11. metadata-eval 99.4%

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

10. Applied egg-rr 99.4%

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

11. Taylor expanded in x around 0 18.8%

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

Alternative 7: 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 87.9%

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

   1. sqr-neg 87.9%

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

   2. remove-double-neg 87.9%

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

   3. distribute-neg-frac 87.9%

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

   4. distribute-frac-neg2 87.9%

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

   5. metadata-eval 87.9%

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

   6. neg-sub0 87.9%

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

   7. associate--r- 87.9%

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

   8. metadata-eval 87.9%

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

   9. +-commutative 87.9%

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

   10. sqr-neg 87.9%

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

   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. Taylor expanded in x around 0 9.5%

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

Reproduce

herbie shell --seed 2024143 
(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))))