ENA, Section 1.4, Mentioned, B

Percentage Accurate: 87.8% → 99.6%

Time: 11.0s

Alternatives: 7

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 87.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.7%

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

   1. sqr-neg 87.7%

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

   2. remove-double-neg 87.7%

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

   3. distribute-neg-frac 87.7%

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

   4. distribute-frac-neg2 87.7%

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

   5. metadata-eval 87.7%

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

   6. neg-sub0 87.7%

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

   7. associate--r- 87.7%

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

   8. metadata-eval 87.7%

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

   9. +-commutative 87.7%

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

   10. sqr-neg 87.7%

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

   11. fma-define 99.5%

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

3. Simplified 99.5%

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

Alternative 2: 88.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

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 + t_0) / (1.0d0 - (x * (x * (x * (x * t_0))))))) * (1.0d0 / (1.0d0 + (x * x))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.7%

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

   1. flip-- 87.6%

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

   2. div-inv 87.6%

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

   3. metadata-eval 87.6%

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

4. Applied egg-rr 87.6%

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

   1. flip-- 87.5%

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

   2. clear-num 87.5%

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

   3. metadata-eval 87.5%

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

6. Applied egg-rr 87.5%

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

   1. expm1-log1p-u 87.3%

      \[\leadsto \frac{10}{\frac{1}{\frac{1 + \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}} \cdot \frac{1}{1 + x \cdot x}} \]

   2. expm1-undefine 87.2%

      \[\leadsto \frac{10}{\frac{1}{\frac{1 + \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} - 1\right)} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}} \cdot \frac{1}{1 + x \cdot x}} \]

   3. log1p-undefine 87.4%

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

   4. add-exp-log 87.4%

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

8. Applied egg-rr 87.4%

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

   1. add-exp-log 87.4%

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

   2. expm1-define 87.3%

      \[\leadsto \frac{10}{\frac{1}{\frac{1 + \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{1 - \color{blue}{\mathsf{expm1}\left(\log \left(1 + \left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}} \cdot \frac{1}{1 + x \cdot x}} \]

   3. log1p-define 87.3%

      \[\leadsto \frac{10}{\frac{1}{\frac{1 + \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{1 - \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}} \cdot \frac{1}{1 + x \cdot x}} \]

   4. add-sqr-sqrt 87.3%

      \[\leadsto \frac{10}{\frac{1}{\frac{1 + \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{1 - \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} \cdot \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}} \cdot \frac{1}{1 + x \cdot x}} \]

   5. sqrt-prod 87.3%

      \[\leadsto \frac{10}{\frac{1}{\frac{1 + \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{1 - \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}} \cdot \frac{1}{1 + x \cdot x}} \]

   6. associate-*r* 87.3%

      \[\leadsto \frac{10}{\frac{1}{\frac{1 + \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{1 - \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}} \cdot \frac{1}{1 + x \cdot x}} \]

   7. expm1-log1p-u 87.5%

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

   8. add-sqr-sqrt 87.5%

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

   9. sqrt-prod 87.5%

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

   10. associate-*r* 87.5%

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

   11. add-sqr-sqrt 87.5%

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

   12. associate-*l* 88.3%

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

10. Applied egg-rr 88.8%

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

Alternative 3: 88.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.7%

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

   1. flip-- 87.6%

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

   2. div-inv 87.6%

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

   3. metadata-eval 87.6%

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

4. Applied egg-rr 87.6%

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

   1. associate-*r* 88.3%

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

   2. associate-*r* 88.3%

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

6. Applied egg-rr 88.3%

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

7. Final simplification 88.3%

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

Alternative 4: 87.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.7%

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

   1. sqr-neg 87.7%

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

   2. remove-double-neg 87.7%

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

   3. distribute-neg-frac 87.7%

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

   4. distribute-frac-neg2 87.7%

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

   5. metadata-eval 87.7%

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

   6. neg-sub0 87.7%

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

   7. associate--r- 87.7%

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

   8. metadata-eval 87.7%

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

   9. +-commutative 87.7%

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

   10. sqr-neg 87.7%

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

   11. fma-define 99.5%

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

3. Simplified 99.5%

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

   1. clear-num 99.4%

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

   2. inv-pow 99.4%

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

   3. add-sqr-sqrt 35.6%

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

   4. associate-/l* 35.6%

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

   5. unpow-prod-down 35.7%

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

   6. inv-pow 35.7%

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

   7. pow1/2 35.7%

      \[\leadsto \frac{1}{\color{blue}{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{0.5}}} \cdot {\left(\frac{\sqrt{\mathsf{fma}\left(x, x, -1\right)}}{-10}\right)}^{-1} \]

   8. fma-undefine 31.8%

      \[\leadsto \frac{1}{{\color{blue}{\left(x \cdot x + -1\right)}}^{0.5}} \cdot {\left(\frac{\sqrt{\mathsf{fma}\left(x, x, -1\right)}}{-10}\right)}^{-1} \]

   9. +-commutative 31.8%

      \[\leadsto \frac{1}{{\color{blue}{\left(-1 + x \cdot x\right)}}^{0.5}} \cdot {\left(\frac{\sqrt{\mathsf{fma}\left(x, x, -1\right)}}{-10}\right)}^{-1} \]

   10. pow1/2 31.8%

       \[\leadsto \frac{1}{{\left(-1 + x \cdot x\right)}^{0.5}} \cdot {\left(\frac{\color{blue}{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{0.5}}}{-10}\right)}^{-1} \]

   11. fma-undefine 31.3%

       \[\leadsto \frac{1}{{\left(-1 + x \cdot x\right)}^{0.5}} \cdot {\left(\frac{{\color{blue}{\left(x \cdot x + -1\right)}}^{0.5}}{-10}\right)}^{-1} \]

   12. +-commutative 31.3%

       \[\leadsto \frac{1}{{\left(-1 + x \cdot x\right)}^{0.5}} \cdot {\left(\frac{{\color{blue}{\left(-1 + x \cdot x\right)}}^{0.5}}{-10}\right)}^{-1} \]

6. Applied egg-rr 31.3%

   \[\leadsto \color{blue}{\frac{1}{{\left(-1 + x \cdot x\right)}^{0.5}} \cdot {\left(\frac{{\left(-1 + x \cdot x\right)}^{0.5}}{-10}\right)}^{-1}} \]
7. Step-by-step derivation

   1. unpow-1 31.3%

      \[\leadsto \frac{1}{{\left(-1 + x \cdot x\right)}^{0.5}} \cdot \color{blue}{\frac{1}{\frac{{\left(-1 + x \cdot x\right)}^{0.5}}{-10}}} \]

   2. associate-*r/ 31.3%

      \[\leadsto \color{blue}{\frac{\frac{1}{{\left(-1 + x \cdot x\right)}^{0.5}} \cdot 1}{\frac{{\left(-1 + x \cdot x\right)}^{0.5}}{-10}}} \]

   3. *-rgt-identity 31.3%

      \[\leadsto \frac{\color{blue}{\frac{1}{{\left(-1 + x \cdot x\right)}^{0.5}}}}{\frac{{\left(-1 + x \cdot x\right)}^{0.5}}{-10}} \]

   4. unpow1/2 31.3%

      \[\leadsto \frac{\frac{1}{\color{blue}{\sqrt{-1 + x \cdot x}}}}{\frac{{\left(-1 + x \cdot x\right)}^{0.5}}{-10}} \]

   5. +-commutative 31.3%

      \[\leadsto \frac{\frac{1}{\sqrt{\color{blue}{x \cdot x + -1}}}}{\frac{{\left(-1 + x \cdot x\right)}^{0.5}}{-10}} \]

   6. fma-define 31.8%

      \[\leadsto \frac{\frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}}}{\frac{{\left(-1 + x \cdot x\right)}^{0.5}}{-10}} \]

   7. unpow1/2 31.8%

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

   8. +-commutative 31.8%

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

   9. fma-define 35.6%

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

8. Simplified 35.6%

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

   1. clear-num 35.6%

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

   2. clear-num 35.6%

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

   3. associate-/r/ 35.6%

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

   4. div-inv 35.6%

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

   5. frac-times 35.6%

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

   6. metadata-eval 35.6%

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

   7. add-sqr-sqrt 99.3%

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

   8. fma-undefine 87.7%

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

10. Applied egg-rr 87.7%

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

11. Final simplification 87.7%

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

Alternative 5: 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. Step-by-step derivation

      1. sqr-neg 88.1%

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

      2. remove-double-neg 88.1%

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

      3. distribute-neg-frac 88.1%

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

      4. distribute-frac-neg2 88.1%

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

      5. metadata-eval 88.1%

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

      6. neg-sub0 88.1%

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

      7. associate--r- 88.1%

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

      8. metadata-eval 88.1%

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

      9. +-commutative 88.1%

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

      10. sqr-neg 88.1%

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

      11. fma-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 13.5%

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

   if 1 < (*.f64 x x)

   1. Initial program 87.0%

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

      1. sqr-neg 87.0%

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

      2. remove-double-neg 87.0%

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

      3. distribute-neg-frac 87.0%

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

      4. distribute-frac-neg2 87.0%

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

      5. metadata-eval 87.0%

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

      6. neg-sub0 87.0%

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

      7. associate--r- 87.0%

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

      8. metadata-eval 87.0%

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

      9. +-commutative 87.0%

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

      10. sqr-neg 87.0%

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

      11. fma-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 13.5%

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

      1. unpow2 13.5%

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

   7. Simplified 13.5%

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

Alternative 6: 87.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.7%

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

Alternative 7: 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 87.7%

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

   1. sqr-neg 87.7%

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

   2. remove-double-neg 87.7%

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

   3. distribute-neg-frac 87.7%

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

   4. distribute-frac-neg2 87.7%

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

   5. metadata-eval 87.7%

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

   6. neg-sub0 87.7%

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

   7. associate--r- 87.7%

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

   8. metadata-eval 87.7%

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

   9. +-commutative 87.7%

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

   10. sqr-neg 87.7%

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

   11. fma-define 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in x around 0 9.2%

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

Reproduce

herbie shell --seed 2024097 
(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))))