ENA, Section 1.4, Mentioned, B

Average Error: 7.8 → 7.8

Time: 28.3s

Precision: binary64

Cost: 3136

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

Math

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

↓

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

FPCore

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

↓

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

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 = t_0 * (1.0 / t_0);
	return 10.0 / (((t_0 * t_1) * t_1) * t_1);
}

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
    t_0 = 1.0d0 - (x * x)
    t_1 = t_0 * (1.0d0 / t_0)
    code = 10.0d0 / (((t_0 * t_1) * t_1) * t_1)
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 = t_0 * (1.0 / t_0);
	return 10.0 / (((t_0 * t_1) * t_1) * t_1);
}

Python

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

↓

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

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(t_0 * Float64(1.0 / t_0))
	return Float64(10.0 / Float64(Float64(Float64(t_0 * t_1) * t_1) * t_1))
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 = t_0 * (1.0 / t_0);
	tmp = 10.0 / (((t_0 * t_1) * t_1) * t_1);
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[(t$95$0 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(10.0 / N[(N[(N[(t$95$0 * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 7.8

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

2. Applied egg-rr 7.8

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

3. Applied egg-rr 7.8

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

4. Applied egg-rr 7.8

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

5. Final simplification 7.8

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

Alternatives

Alternative 1
Error	7.8
Cost	2240

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

Alternative 2
Error	7.8
Cost	1344

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

Alternative 3
Error	7.8
Cost	1344

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

Alternative 4
Error	7.8
Cost	448

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

Alternative 5
Error	57.9
Cost	64

\[10 \]

Reproduce

herbie shell --seed 2023090 
(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))))