ENA, Section 1.4, Mentioned, B

Percentage Accurate: 87.8% → 99.6%

Time: 10.6s

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: 87.8% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{10}{\mathsf{fma}\left(0 - x, x, 1\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 10.0 (fma (- 0.0 x) x 1.0)))

C

double code(double x) {
	return 10.0 / fma((0.0 - x), x, 1.0);
}

Julia

function code(x)
	return Float64(10.0 / fma(Float64(0.0 - x), x, 1.0))
end

Wolfram

code[x_] := N[(10.0 / N[(N[(0.0 - x), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.0%

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

   1. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(1 + \color{blue}{\left(\mathsf{neg}\left(x \cdot x\right)\right)}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(\left(\mathsf{neg}\left(x \cdot x\right)\right) + \color{blue}{1}\right)\right) \]

   3. distribute-lft-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot x + 1\right)\right) \]

   4. fma-define N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(\mathsf{fma}\left(\mathsf{neg}\left(x\right), \color{blue}{x}, 1\right)\right)\right) \]

   5. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(10, \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(x\right)\right), \color{blue}{x}, 1\right)\right) \]

   6. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(10, \mathsf{fma.f64}\left(\left(0 - x\right), x, 1\right)\right) \]

   7. --lowering--.f64 99.7%

      \[\leadsto \mathsf{/.f64}\left(10, \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, x\right), x, 1\right)\right) \]

4. Applied egg-rr 99.7%

   \[\leadsto \frac{10}{\color{blue}{\mathsf{fma}\left(0 - x, x, 1\right)}} \]
5. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{/.f64}\left(10, \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(x\right)\right), x, 1\right)\right) \]

   2. neg-lowering-neg.f64 99.7%

      \[\leadsto \mathsf{/.f64}\left(10, \mathsf{fma.f64}\left(\mathsf{neg.f64}\left(x\right), x, 1\right)\right) \]

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

   \[\leadsto \frac{10}{\mathsf{fma}\left(0 - x, x, 1\right)} \]
8. Add Preprocessing

Alternative 2: 13.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1:\\ \;\;\;\;10\\ \mathbf{else}:\\ \;\;\;\;\frac{-10}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (* x x) 1.0) 10.0 (/ -10.0 (* x x))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 1.0) {
		tmp = 10.0;
	} else {
		tmp = -10.0 / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x * x) <= 1.0d0) then
        tmp = 10.0d0
    else
        tmp = (-10.0d0) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 1.0) {
		tmp = 10.0;
	} else {
		tmp = -10.0 / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 1.0:
		tmp = 10.0
	else:
		tmp = -10.0 / (x * x)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 1.0)
		tmp = 10.0;
	else
		tmp = -10.0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1

   1. Initial program 88.4%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 13.5%

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

   if 1 < (*.f64 x x)

   1. Initial program 87.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-10}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-10, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(-10, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 13.5%

         \[\leadsto \mathsf{/.f64}\left(-10, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   5. Simplified 13.5%

      \[\leadsto \color{blue}{\frac{-10}{x \cdot x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 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 88.0%

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

   1. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(1 + \color{blue}{\left(\mathsf{neg}\left(x \cdot x\right)\right)}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(\left(\mathsf{neg}\left(x \cdot x\right)\right) + \color{blue}{1}\right)\right) \]

   3. distribute-lft-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot x + 1\right)\right) \]

   4. fma-define N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(\mathsf{fma}\left(\mathsf{neg}\left(x\right), \color{blue}{x}, 1\right)\right)\right) \]

   5. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(10, \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(x\right)\right), \color{blue}{x}, 1\right)\right) \]

   6. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(10, \mathsf{fma.f64}\left(\left(0 - x\right), x, 1\right)\right) \]

   7. --lowering--.f64 99.7%

      \[\leadsto \mathsf{/.f64}\left(10, \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, x\right), x, 1\right)\right) \]

4. Applied egg-rr 99.7%

   \[\leadsto \frac{10}{\color{blue}{\mathsf{fma}\left(0 - x, x, 1\right)}} \]
5. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot x + 1\right)\right) \]

   2. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(\left(\mathsf{neg}\left(x \cdot x\right)\right) + 1\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(\left(\mathsf{neg}\left(x \cdot x\right)\right) + \left(\mathsf{neg}\left(-1\right)\right)\right)\right) \]

   4. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(\mathsf{neg}\left(\left(x \cdot x + -1\right)\right)\right)\right) \]

   5. difference-of-sqr--1 N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(\mathsf{neg}\left(\left(x + 1\right) \cdot \left(x - 1\right)\right)\right)\right) \]

   6. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(\left(x + 1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(x - 1\right)\right)\right)}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(10, \mathsf{*.f64}\left(\left(x + 1\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - 1\right)\right)\right)}\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(10, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - 1\right)}\right)\right)\right)\right) \]

   9. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(10, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{neg.f64}\left(\left(x - 1\right)\right)\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(10, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{neg.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(10, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{neg.f64}\left(\left(x + -1\right)\right)\right)\right) \]

   12. +-lowering-+.f64 99.5%

       \[\leadsto \mathsf{/.f64}\left(10, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{neg.f64}\left(\mathsf{+.f64}\left(x, -1\right)\right)\right)\right) \]

6. Applied egg-rr 99.5%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(\left(\mathsf{neg}\left(\left(x + -1\right)\right)\right) \cdot \color{blue}{\left(x + 1\right)}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(\left(\mathsf{neg}\left(\left(x + -1\right)\right)\right) \cdot \left(1 + \color{blue}{x}\right)\right)\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(1 \cdot \left(\mathsf{neg}\left(\left(x + -1\right)\right)\right) + \color{blue}{x \cdot \left(\mathsf{neg}\left(\left(x + -1\right)\right)\right)}\right)\right) \]

8. Applied egg-rr 99.6%

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

Alternative 4: 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 88.0%

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

   1. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(1 + \color{blue}{\left(\mathsf{neg}\left(x \cdot x\right)\right)}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(\left(\mathsf{neg}\left(x \cdot x\right)\right) + \color{blue}{1}\right)\right) \]

   3. distribute-lft-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot x + 1\right)\right) \]

   4. fma-define N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(\mathsf{fma}\left(\mathsf{neg}\left(x\right), \color{blue}{x}, 1\right)\right)\right) \]

   5. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(10, \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(x\right)\right), \color{blue}{x}, 1\right)\right) \]

   6. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(10, \mathsf{fma.f64}\left(\left(0 - x\right), x, 1\right)\right) \]

   7. --lowering--.f64 99.7%

      \[\leadsto \mathsf{/.f64}\left(10, \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, x\right), x, 1\right)\right) \]

4. Applied egg-rr 99.7%

   \[\leadsto \frac{10}{\color{blue}{\mathsf{fma}\left(0 - x, x, 1\right)}} \]
5. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(\left(\mathsf{neg}\left(x\right)\right) \cdot x + 1\right)\right) \]

   2. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(\left(\mathsf{neg}\left(x \cdot x\right)\right) + 1\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(\left(\mathsf{neg}\left(x \cdot x\right)\right) + \left(\mathsf{neg}\left(-1\right)\right)\right)\right) \]

   4. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(\mathsf{neg}\left(\left(x \cdot x + -1\right)\right)\right)\right) \]

   5. difference-of-sqr--1 N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(\mathsf{neg}\left(\left(x + 1\right) \cdot \left(x - 1\right)\right)\right)\right) \]

   6. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(\left(x + 1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(x - 1\right)\right)\right)}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(10, \mathsf{*.f64}\left(\left(x + 1\right), \color{blue}{\left(\mathsf{neg}\left(\left(x - 1\right)\right)\right)}\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(10, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(\mathsf{neg}\left(\color{blue}{\left(x - 1\right)}\right)\right)\right)\right) \]

   9. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(10, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{neg.f64}\left(\left(x - 1\right)\right)\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(10, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{neg.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(10, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{neg.f64}\left(\left(x + -1\right)\right)\right)\right) \]

   12. +-lowering-+.f64 99.5%

       \[\leadsto \mathsf{/.f64}\left(10, \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{neg.f64}\left(\mathsf{+.f64}\left(x, -1\right)\right)\right)\right) \]

6. Applied egg-rr 99.5%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(\left(\mathsf{neg}\left(\left(x + -1\right)\right)\right) \cdot \color{blue}{\left(x + 1\right)}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(10, \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\left(x + -1\right)\right)\right), \color{blue}{\left(x + 1\right)}\right)\right) \]

   3. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(10, \mathsf{*.f64}\left(\left(0 - \left(x + -1\right)\right), \left(\color{blue}{x} + 1\right)\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(10, \mathsf{*.f64}\left(\left(0 - \left(-1 + x\right)\right), \left(x + 1\right)\right)\right) \]

   5. associate--r+ N/A

      \[\leadsto \mathsf{/.f64}\left(10, \mathsf{*.f64}\left(\left(\left(0 - -1\right) - x\right), \left(\color{blue}{x} + 1\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(10, \mathsf{*.f64}\left(\left(1 - x\right), \left(x + 1\right)\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(10, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \left(\color{blue}{x} + 1\right)\right)\right) \]

   8. +-lowering-+.f64 99.5%

      \[\leadsto \mathsf{/.f64}\left(10, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, x\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

8. Applied egg-rr 99.5%

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

Alternative 5: 87.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

4. Applied egg-rr 88.0%

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

Alternative 6: 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]

Derivation

1. Initial program 88.0%

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

Alternative 7: 9.6% 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 88.0%

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

3. Taylor expanded in x around 0

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

5. Simplified 9.5%

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

Reproduce

herbie shell --seed 2024150 
(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))))