ENA, Section 1.4, Mentioned, B

Percentage Accurate: 87.8% → 99.6%

Time: 21.0s

Alternatives: 12

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 87.9%

   \[\frac{10}{1 - x \cdot x} \]
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. accelerator-lowering-fma.f64 N/A

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

   5. neg-sub0 N/A

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

   6. --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: 88.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + x \cdot x\\ t_1 := x \cdot \left(x \cdot x\right)\\ t_2 := x \cdot t\_1\\ t_3 := t\_0 \cdot \left(1 + t\_2\right)\\ t_4 := \frac{t\_3}{x \cdot \left(x \cdot \left(t\_1 \cdot t\_1\right)\right)}\\ \frac{10}{\frac{t\_4 + t\_0 \cdot \left(-1 - t\_2\right)}{t\_3 \cdot t\_4}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x x)))
        (t_1 (* x (* x x)))
        (t_2 (* x t_1))
        (t_3 (* t_0 (+ 1.0 t_2)))
        (t_4 (/ t_3 (* x (* x (* t_1 t_1))))))
   (/ 10.0 (/ (+ t_4 (* t_0 (- -1.0 t_2))) (* t_3 t_4)))))

C

double code(double x) {
	double t_0 = 1.0 + (x * x);
	double t_1 = x * (x * x);
	double t_2 = x * t_1;
	double t_3 = t_0 * (1.0 + t_2);
	double t_4 = t_3 / (x * (x * (t_1 * t_1)));
	return 10.0 / ((t_4 + (t_0 * (-1.0 - t_2))) / (t_3 * t_4));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    t_0 = 1.0d0 + (x * x)
    t_1 = x * (x * x)
    t_2 = x * t_1
    t_3 = t_0 * (1.0d0 + t_2)
    t_4 = t_3 / (x * (x * (t_1 * t_1)))
    code = 10.0d0 / ((t_4 + (t_0 * ((-1.0d0) - t_2))) / (t_3 * t_4))
end function

Java

public static double code(double x) {
	double t_0 = 1.0 + (x * x);
	double t_1 = x * (x * x);
	double t_2 = x * t_1;
	double t_3 = t_0 * (1.0 + t_2);
	double t_4 = t_3 / (x * (x * (t_1 * t_1)));
	return 10.0 / ((t_4 + (t_0 * (-1.0 - t_2))) / (t_3 * t_4));
}

Python

def code(x):
	t_0 = 1.0 + (x * x)
	t_1 = x * (x * x)
	t_2 = x * t_1
	t_3 = t_0 * (1.0 + t_2)
	t_4 = t_3 / (x * (x * (t_1 * t_1)))
	return 10.0 / ((t_4 + (t_0 * (-1.0 - t_2))) / (t_3 * t_4))

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(x * x))
	t_1 = Float64(x * Float64(x * x))
	t_2 = Float64(x * t_1)
	t_3 = Float64(t_0 * Float64(1.0 + t_2))
	t_4 = Float64(t_3 / Float64(x * Float64(x * Float64(t_1 * t_1))))
	return Float64(10.0 / Float64(Float64(t_4 + Float64(t_0 * Float64(-1.0 - t_2))) / Float64(t_3 * t_4)))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 + (x * x);
	t_1 = x * (x * x);
	t_2 = x * t_1;
	t_3 = t_0 * (1.0 + t_2);
	t_4 = t_3 / (x * (x * (t_1 * t_1)));
	tmp = 10.0 / ((t_4 + (t_0 * (-1.0 - t_2))) / (t_3 * t_4));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / N[(x * N[(x * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(10.0 / N[(N[(t$95$4 + N[(t$95$0 * N[(-1.0 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 87.9%

   \[\frac{10}{1 - x \cdot x} \]
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. accelerator-lowering-fma.f64 N/A

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

   5. neg-sub0 N/A

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

   6. --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%

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

8. Applied egg-rr 88.9%

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

9. Final simplification 88.9%

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

Alternative 3: 88.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + x \cdot x\\ t_1 := x \cdot \left(x \cdot x\right)\\ t_2 := x \cdot \left(x \cdot \left(t\_1 \cdot t\_1\right)\right)\\ \frac{10}{\frac{\frac{1 - t\_2 \cdot t\_2}{t\_0 \cdot t\_0} \cdot \frac{1}{\frac{1 + t\_2}{t\_0}}}{1 + x \cdot t\_1}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x x)))
        (t_1 (* x (* x x)))
        (t_2 (* x (* x (* t_1 t_1)))))
   (/
    10.0
    (/
     (* (/ (- 1.0 (* t_2 t_2)) (* t_0 t_0)) (/ 1.0 (/ (+ 1.0 t_2) t_0)))
     (+ 1.0 (* x t_1))))))

C

double code(double x) {
	double t_0 = 1.0 + (x * x);
	double t_1 = x * (x * x);
	double t_2 = x * (x * (t_1 * t_1));
	return 10.0 / ((((1.0 - (t_2 * t_2)) / (t_0 * t_0)) * (1.0 / ((1.0 + t_2) / t_0))) / (1.0 + (x * t_1)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = 1.0d0 + (x * x)
    t_1 = x * (x * x)
    t_2 = x * (x * (t_1 * t_1))
    code = 10.0d0 / ((((1.0d0 - (t_2 * t_2)) / (t_0 * t_0)) * (1.0d0 / ((1.0d0 + t_2) / t_0))) / (1.0d0 + (x * t_1)))
end function

Java

public static double code(double x) {
	double t_0 = 1.0 + (x * x);
	double t_1 = x * (x * x);
	double t_2 = x * (x * (t_1 * t_1));
	return 10.0 / ((((1.0 - (t_2 * t_2)) / (t_0 * t_0)) * (1.0 / ((1.0 + t_2) / t_0))) / (1.0 + (x * t_1)));
}

Python

def code(x):
	t_0 = 1.0 + (x * x)
	t_1 = x * (x * x)
	t_2 = x * (x * (t_1 * t_1))
	return 10.0 / ((((1.0 - (t_2 * t_2)) / (t_0 * t_0)) * (1.0 / ((1.0 + t_2) / t_0))) / (1.0 + (x * t_1)))

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(x * x))
	t_1 = Float64(x * Float64(x * x))
	t_2 = Float64(x * Float64(x * Float64(t_1 * t_1)))
	return Float64(10.0 / Float64(Float64(Float64(Float64(1.0 - Float64(t_2 * t_2)) / Float64(t_0 * t_0)) * Float64(1.0 / Float64(Float64(1.0 + t_2) / t_0))) / Float64(1.0 + Float64(x * t_1))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 + (x * x);
	t_1 = x * (x * x);
	t_2 = x * (x * (t_1 * t_1));
	tmp = 10.0 / ((((1.0 - (t_2 * t_2)) / (t_0 * t_0)) * (1.0 / ((1.0 + t_2) / t_0))) / (1.0 + (x * t_1)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(x * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(10.0 / N[(N[(N[(N[(1.0 - N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(1.0 + t$95$2), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 87.9%

   \[\frac{10}{1 - x \cdot x} \]
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. accelerator-lowering-fma.f64 N/A

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

   5. neg-sub0 N/A

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

   6. --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. flip-- N/A

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

   8. metadata-eval N/A

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

   9. associate-*r* N/A

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

   10. associate-*r* N/A

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

   11. associate-/r* N/A

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

   12. *-commutative N/A

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

6. Applied egg-rr 88.1%

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

8. Applied egg-rr 88.8%

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

Alternative 4: 88.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := x \cdot \left(x \cdot \left(t\_0 \cdot t\_0\right)\right)\\ t_2 := 1 + x \cdot x\\ \frac{10}{\frac{\frac{1 - t\_1 \cdot t\_1}{t\_2 \cdot t\_2}}{\left(1 + x \cdot t\_0\right) \cdot \frac{1 + t\_1}{t\_2}}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x)))
        (t_1 (* x (* x (* t_0 t_0))))
        (t_2 (+ 1.0 (* x x))))
   (/
    10.0
    (/
     (/ (- 1.0 (* t_1 t_1)) (* t_2 t_2))
     (* (+ 1.0 (* x t_0)) (/ (+ 1.0 t_1) t_2))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = x * (x * (t_0 * t_0));
	double t_2 = 1.0 + (x * x);
	return 10.0 / (((1.0 - (t_1 * t_1)) / (t_2 * t_2)) / ((1.0 + (x * t_0)) * ((1.0 + t_1) / t_2)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = x * (x * x)
    t_1 = x * (x * (t_0 * t_0))
    t_2 = 1.0d0 + (x * x)
    code = 10.0d0 / (((1.0d0 - (t_1 * t_1)) / (t_2 * t_2)) / ((1.0d0 + (x * t_0)) * ((1.0d0 + t_1) / t_2)))
end function

Java

public static double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = x * (x * (t_0 * t_0));
	double t_2 = 1.0 + (x * x);
	return 10.0 / (((1.0 - (t_1 * t_1)) / (t_2 * t_2)) / ((1.0 + (x * t_0)) * ((1.0 + t_1) / t_2)));
}

Python

def code(x):
	t_0 = x * (x * x)
	t_1 = x * (x * (t_0 * t_0))
	t_2 = 1.0 + (x * x)
	return 10.0 / (((1.0 - (t_1 * t_1)) / (t_2 * t_2)) / ((1.0 + (x * t_0)) * ((1.0 + t_1) / t_2)))

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(x * Float64(x * Float64(t_0 * t_0)))
	t_2 = Float64(1.0 + Float64(x * x))
	return Float64(10.0 / Float64(Float64(Float64(1.0 - Float64(t_1 * t_1)) / Float64(t_2 * t_2)) / Float64(Float64(1.0 + Float64(x * t_0)) * Float64(Float64(1.0 + t_1) / t_2))))
end

MATLAB

function tmp = code(x)
	t_0 = x * (x * x);
	t_1 = x * (x * (t_0 * t_0));
	t_2 = 1.0 + (x * x);
	tmp = 10.0 / (((1.0 - (t_1 * t_1)) / (t_2 * t_2)) / ((1.0 + (x * t_0)) * ((1.0 + t_1) / t_2)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[(10.0 / N[(N[(N[(1.0 - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 87.9%

   \[\frac{10}{1 - x \cdot x} \]
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. accelerator-lowering-fma.f64 N/A

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

   5. neg-sub0 N/A

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

   6. --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%

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

8. Applied egg-rr 88.8%

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

Alternative 5: 88.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.9%

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

4. Applied egg-rr 88.3%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

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

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

   7. *-lowering-*.f64 88.8%

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

6. Applied egg-rr 88.8%

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

7. Final simplification 88.8%

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

Alternative 6: 88.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.9%

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

4. Applied egg-rr 88.3%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

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

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

   7. *-lowering-*.f64 88.8%

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

6. Applied egg-rr 88.8%

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

7. Final simplification 88.8%

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

Alternative 7: 88.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.9%

   \[\frac{10}{1 - x \cdot x} \]
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. *-lowering-*.f64 N/A

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

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

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

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

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

   10. metadata-eval N/A

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

   11. sub-neg N/A

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

   12. distribute-neg-in N/A

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

   13. metadata-eval N/A

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

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

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

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

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

   16. sqr-neg N/A

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

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

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

   18. associate-*l* N/A

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

4. Applied egg-rr 88.6%

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

5. Final simplification 88.6%

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

Alternative 8: 88.5% 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 87.9%

   \[\frac{10}{1 - x \cdot x} \]
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. accelerator-lowering-fma.f64 N/A

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

   5. neg-sub0 N/A

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

   6. --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 88.6%

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

6. Applied egg-rr 88.6%

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

Alternative 9: 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 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 10: 87.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.9%

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

4. Applied egg-rr 88.3%

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

6. Applied egg-rr 88.3%

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

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

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

   2. un-div-inv N/A

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

   3. associate-*r/ N/A

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

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

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

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

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

   6. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

   11. *-lowering-*.f64 88.3%

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

8. Applied egg-rr 88.3%

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

10. Applied egg-rr 88.0%

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

Alternative 11: 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 87.9%

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

Alternative 12: 9.4% 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. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Simplified 9.8%

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

Reproduce

herbie shell --seed 2024192 
(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))))