ENA, Section 1.4, Mentioned, B

Average Error: 7.8 → 7.8

Time: 58.3s

Precision: binary64

Cost: 5824

\[0.999 \leq x \land x \leq 1.001\]

Math

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

↓

\[\begin{array}{l} t_0 := 1 - x \cdot x\\ t_1 := \frac{10}{t_0}\\ t_2 := \frac{10}{\frac{1}{t_0} \cdot \left(t_0 \cdot t_0\right)}\\ \frac{\frac{1}{t_1}}{t_2 \cdot t_2} \cdot \left(\left(0 - \left(-1 - t_1 \cdot \left(t_1 \cdot \left(t_1 \cdot t_1\right)\right)\right)\right) - 1\right) \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 1.0 (* x x)))
        (t_1 (/ 10.0 t_0))
        (t_2 (/ 10.0 (* (/ 1.0 t_0) (* t_0 t_0)))))
   (*
    (/ (/ 1.0 t_1) (* t_2 t_2))
    (- (- 0.0 (- -1.0 (* t_1 (* t_1 (* t_1 t_1))))) 1.0))))

C

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

↓

double code(double x) {
	double t_0 = 1.0 - (x * x);
	double t_1 = 10.0 / t_0;
	double t_2 = 10.0 / ((1.0 / t_0) * (t_0 * t_0));
	return ((1.0 / t_1) / (t_2 * t_2)) * ((0.0 - (-1.0 - (t_1 * (t_1 * (t_1 * t_1))))) - 1.0);
}

Fortran

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

↓

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 = 10.0d0 / t_0
    t_2 = 10.0d0 / ((1.0d0 / t_0) * (t_0 * t_0))
    code = ((1.0d0 / t_1) / (t_2 * t_2)) * ((0.0d0 - ((-1.0d0) - (t_1 * (t_1 * (t_1 * t_1))))) - 1.0d0)
end function

Java

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

↓

public static double code(double x) {
	double t_0 = 1.0 - (x * x);
	double t_1 = 10.0 / t_0;
	double t_2 = 10.0 / ((1.0 / t_0) * (t_0 * t_0));
	return ((1.0 / t_1) / (t_2 * t_2)) * ((0.0 - (-1.0 - (t_1 * (t_1 * (t_1 * t_1))))) - 1.0);
}

Python

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

↓

def code(x):
	t_0 = 1.0 - (x * x)
	t_1 = 10.0 / t_0
	t_2 = 10.0 / ((1.0 / t_0) * (t_0 * t_0))
	return ((1.0 / t_1) / (t_2 * t_2)) * ((0.0 - (-1.0 - (t_1 * (t_1 * (t_1 * t_1))))) - 1.0)

Julia

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

↓

function code(x)
	t_0 = Float64(1.0 - Float64(x * x))
	t_1 = Float64(10.0 / t_0)
	t_2 = Float64(10.0 / Float64(Float64(1.0 / t_0) * Float64(t_0 * t_0)))
	return Float64(Float64(Float64(1.0 / t_1) / Float64(t_2 * t_2)) * Float64(Float64(0.0 - Float64(-1.0 - Float64(t_1 * Float64(t_1 * Float64(t_1 * t_1))))) - 1.0))
end

MATLAB

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

↓

function tmp = code(x)
	t_0 = 1.0 - (x * x);
	t_1 = 10.0 / t_0;
	t_2 = 10.0 / ((1.0 / t_0) * (t_0 * t_0));
	tmp = ((1.0 / t_1) / (t_2 * t_2)) * ((0.0 - (-1.0 - (t_1 * (t_1 * (t_1 * t_1))))) - 1.0);
end

Wolfram

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

↓

code[x_] := Block[{t$95$0 = N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(10.0 / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(10.0 / N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 / t$95$1), $MachinePrecision] / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[(0.0 - N[(-1.0 - N[(t$95$1 * N[(t$95$1 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 7.8

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

2. Applied egg-rr 7.8

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

3. Applied egg-rr 7.8

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

4. Applied egg-rr 7.8

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

5. Applied egg-rr 7.8

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

6. Final simplification 7.8

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

Alternatives

Alternative 1
Error	7.8
Cost	5824

\[\begin{array}{l} t_0 := 1 - x \cdot x\\ t_1 := \frac{10}{\frac{1}{t_0} \cdot \left(t_0 \cdot t_0\right)}\\ t_2 := \frac{10}{t_0}\\ \frac{\frac{1}{t_1}}{t_2 \cdot t_1} \cdot \left(\left(0 - \left(-1 - t_2 \cdot \left(t_2 \cdot \left(t_2 \cdot t_2\right)\right)\right)\right) - 1\right) \end{array} \]

Alternative 2
Error	7.8
Cost	4928

\[\begin{array}{l} t_0 := 1 - x \cdot x\\ t_1 := \frac{10}{t_0}\\ \frac{\frac{1}{t_1}}{t_1 \cdot \frac{10}{\frac{1}{t_0} \cdot \left(t_0 \cdot t_0\right)}} \cdot \left(\left(0 - \left(-1 - t_1 \cdot \left(t_1 \cdot \left(t_1 \cdot t_1\right)\right)\right)\right) - 1\right) \end{array} \]

Alternative 3
Error	7.8
Cost	4032

\[\begin{array}{l} t_0 := \frac{10}{1 - x \cdot x}\\ t_1 := t_0 \cdot t_0\\ \frac{\frac{1}{t_0}}{t_1} \cdot \left(\left(0 - \left(-1 - t_0 \cdot \left(t_0 \cdot t_1\right)\right)\right) - 1\right) \end{array} \]

Alternative 4
Error	7.8
Cost	3648

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

Alternative 5
Error	7.8
Cost	3648

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

Alternative 6
Error	7.8
Cost	2496

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

Alternative 7
Error	7.8
Cost	2240

\[\begin{array}{l} t_0 := 1 - x \cdot x\\ t_1 := t_0 \cdot t_0\\ \frac{10}{\frac{1}{\frac{1}{t_0} \cdot t_1} \cdot t_1} \end{array} \]

Alternative 8
Error	7.8
Cost	448

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

Alternative 9
Error	57.9
Cost	64

\[10 \]

Reproduce

herbie shell --seed 2023077 
(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))))