ENA, Section 1.4, Mentioned, B

Percentage Accurate: 87.8% → 99.6%

Time: 11.9s

Alternatives: 8

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 87.8%

   \[\frac{10}{1 - x \cdot x} \]
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 99.6%

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

4. Applied egg-rr 99.6%

   \[\leadsto \frac{10}{\color{blue}{\mathsf{fma}\left(0 - x, x, 1\right)}} \]
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 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 88.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := x \cdot t\_0\\ \frac{10}{\frac{1 - \left(x \cdot \left(t\_0 \cdot t\_0\right)\right) \cdot \left(x \cdot t\_1\right)}{\left(1 + x \cdot x\right) \cdot \left(1 + t\_1 \cdot \left(1 + t\_1\right)\right)}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))) (t_1 (* x t_0)))
   (/
    10.0
    (/
     (- 1.0 (* (* x (* t_0 t_0)) (* x t_1)))
     (* (+ 1.0 (* x x)) (+ 1.0 (* t_1 (+ 1.0 t_1))))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = x * t_0;
	return 10.0 / ((1.0 - ((x * (t_0 * t_0)) * (x * t_1))) / ((1.0 + (x * x)) * (1.0 + (t_1 * (1.0 + t_1)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    t_0 = x * (x * x)
    t_1 = x * t_0
    code = 10.0d0 / ((1.0d0 - ((x * (t_0 * t_0)) * (x * t_1))) / ((1.0d0 + (x * x)) * (1.0d0 + (t_1 * (1.0d0 + t_1)))))
end function

Java

public static double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = x * t_0;
	return 10.0 / ((1.0 - ((x * (t_0 * t_0)) * (x * t_1))) / ((1.0 + (x * x)) * (1.0 + (t_1 * (1.0 + t_1)))));
}

Python

def code(x):
	t_0 = x * (x * x)
	t_1 = x * t_0
	return 10.0 / ((1.0 - ((x * (t_0 * t_0)) * (x * t_1))) / ((1.0 + (x * x)) * (1.0 + (t_1 * (1.0 + t_1)))))

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(x * t_0)
	return Float64(10.0 / Float64(Float64(1.0 - Float64(Float64(x * Float64(t_0 * t_0)) * Float64(x * t_1))) / Float64(Float64(1.0 + Float64(x * x)) * Float64(1.0 + Float64(t_1 * Float64(1.0 + t_1))))))
end

MATLAB

function tmp = code(x)
	t_0 = x * (x * x);
	t_1 = x * t_0;
	tmp = 10.0 / ((1.0 - ((x * (t_0 * t_0)) * (x * t_1))) / ((1.0 + (x * x)) * (1.0 + (t_1 * (1.0 + t_1)))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, N[(10.0 / N[(N[(1.0 - N[(N[(x * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(t$95$1 * N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 87.8%

   \[\frac{10}{1 - x \cdot x} \]
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 99.6%

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

4. Applied egg-rr 99.6%

   \[\leadsto \frac{10}{\color{blue}{\mathsf{fma}\left(0 - x, x, 1\right)}} \]
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 99.6%

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

6. Applied egg-rr 99.6%

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

   1. +-commutative N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. flip-- N/A

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

   4. metadata-eval N/A

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

   5. associate-*r* N/A

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

   6. flip3-- N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(\frac{\frac{{1}^{3} - {\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}^{3}}{1 \cdot 1 + \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + 1 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}}{\color{blue}{1} + x \cdot x}\right)\right) \]

   7. associate-/l/ N/A

      \[\leadsto \mathsf{/.f64}\left(10, \left(\frac{{1}^{3} - {\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}^{3}}{\color{blue}{\left(1 + x \cdot x\right) \cdot \left(1 \cdot 1 + \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + 1 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(10, \mathsf{/.f64}\left(\left({1}^{3} - {\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}^{3}\right), \color{blue}{\left(\left(1 + x \cdot x\right) \cdot \left(1 \cdot 1 + \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) + 1 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)}\right)\right) \]

8. Applied egg-rr 88.4%

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

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

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

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

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

   6. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

   12. associate-*r* N/A

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

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

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

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

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

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

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

   16. *-lowering-*.f64 88.9%

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

10. Applied egg-rr 88.9%

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

Alternative 3: 88.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

   1. flip-- N/A

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

   2. associate-/r/ N/A

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

   3. associate-*l/ N/A

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

   4. frac-2neg N/A

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

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

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

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

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

   7. distribute-lft-in N/A

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

   8. metadata-eval N/A

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

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

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

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

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

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

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

   12. metadata-eval N/A

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

   13. sub-neg N/A

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

   14. distribute-neg-in N/A

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

   15. metadata-eval N/A

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

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

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

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

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

   18. sqr-neg N/A

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

4. Applied egg-rr 88.8%

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

   1. neg-sub0 N/A

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

   2. +-commutative N/A

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

   3. associate--r+ N/A

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

   4. neg-sub0 N/A

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

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

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

   6. *-commutative N/A

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

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

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

   8. metadata-eval N/A

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

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

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

   10. *-lowering-*.f64 88.8%

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

6. Applied egg-rr 88.8%

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

7. Final simplification 88.8%

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

Alternative 4: 88.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

   \[\frac{10}{1 - x \cdot x} \]
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 99.6%

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

4. Applied egg-rr 99.6%

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

   1. +-commutative N/A

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

   2. sub0-neg N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. flip-- N/A

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

   5. metadata-eval N/A

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

   6. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   13. *-lowering-*.f64 88.8%

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

6. Applied egg-rr 88.8%

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

Alternative 5: 88.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

   \[\frac{10}{1 - x \cdot x} \]
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 99.6%

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

4. Applied egg-rr 99.6%

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

   1. +-commutative N/A

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

   2. sub0-neg N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. flip-- N/A

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

   5. metadata-eval N/A

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

   6. associate-*r* N/A

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

   7. associate-/r/ N/A

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

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

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

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

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

   10. --lowering--.f64 N/A

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

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

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

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

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

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

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

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

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

   15. *-lowering-*.f64 88.8%

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

6. Applied egg-rr 88.8%

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

7. Final simplification 88.8%

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

Alternative 6: 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 88.1%

      \[\frac{10}{1 - x \cdot x} \]
   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 86.8%

      \[\frac{10}{1 - x \cdot x} \]
   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 7: 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 87.8%

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

Alternative 8: 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 87.8%

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

3. Taylor expanded in x around 0

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

5. Simplified 10.3%

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

Reproduce

herbie shell --seed 2024158 
(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))))