ENA, Section 1.4, Mentioned, B

Percentage Accurate: 87.8% → 99.6%

Time: 9.1s

Alternatives: 4

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 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-frac2 88.1%

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

   4. distribute-neg-frac 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. Final simplification 99.5%

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

Alternative 2: 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 88.1%

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

3. Final simplification 88.1%

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

Alternative 3: 13.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;10\\ \mathbf{else}:\\ \;\;\;\;-10\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.0) 10.0 -10.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 10.0;
	} else {
		tmp = -10.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = 10.0d0
    else
        tmp = -10.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 10.0;
	} else {
		tmp = -10.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = 10.0
	else:
		tmp = -10.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = 10.0;
	else
		tmp = -10.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 10.0;
	else
		tmp = -10.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], 10.0, -10.0]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 88.4%

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

      1. sqr-neg 88.4%

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

      2. remove-double-neg 88.4%

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

      3. distribute-neg-frac2 88.4%

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

      4. distribute-neg-frac 88.4%

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

      5. metadata-eval 88.4%

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

      6. neg-sub0 88.4%

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

      7. associate--r- 88.4%

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

      8. metadata-eval 88.4%

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

      9. +-commutative 88.4%

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

      10. sqr-neg 88.4%

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

      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 < x

   1. Initial program 87.2%

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

      1. sqr-neg 87.2%

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

      2. remove-double-neg 87.2%

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

      3. distribute-neg-frac2 87.2%

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

      4. distribute-neg-frac 87.2%

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

      5. metadata-eval 87.2%

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

      6. neg-sub0 87.2%

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

      7. associate--r- 87.2%

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

      8. metadata-eval 87.2%

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

      9. +-commutative 87.2%

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

      10. sqr-neg 87.2%

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

      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. frac-2neg 99.5%

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

      2. metadata-eval 99.5%

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

      3. fma-undefine 87.2%

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

      4. distribute-neg-in 87.2%

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

      5. metadata-eval 87.2%

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

      6. +-commutative 87.2%

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

      7. sub-neg 87.2%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 1.5%

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

      10. pow1/2 1.5%

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

   6. Applied egg-rr 1.5%

      \[\leadsto \color{blue}{{\left(\frac{100}{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{2}}\right)}^{0.5}} \]
   7. Step-by-step derivation

      1. unpow1/2 1.5%

         \[\leadsto \color{blue}{\sqrt{\frac{100}{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{2}}}} \]

   8. Simplified 1.5%

      \[\leadsto \color{blue}{\sqrt{\frac{100}{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{2}}}} \]

   9. Taylor expanded in x around 0 13.5%

      \[\leadsto \color{blue}{-10} \]
3. Recombined 2 regimes into one program.

4. Final simplification 13.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;10\\ \mathbf{else}:\\ \;\;\;\;-10\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 5.5% 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 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-frac2 88.1%

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

   4. distribute-neg-frac 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. Step-by-step derivation

   1. frac-2neg 99.5%

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

   2. metadata-eval 99.5%

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

   3. fma-undefine 88.1%

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

   4. distribute-neg-in 88.1%

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

   5. metadata-eval 88.1%

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

   6. +-commutative 88.1%

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

   7. sub-neg 88.1%

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

   8. add-sqr-sqrt 62.2%

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

   9. sqrt-unprod 62.6%

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

   10. pow1/2 62.6%

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

6. Applied egg-rr 70.4%

   \[\leadsto \color{blue}{{\left(\frac{100}{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{2}}\right)}^{0.5}} \]
7. Step-by-step derivation

   1. unpow1/2 70.4%

      \[\leadsto \color{blue}{\sqrt{\frac{100}{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{2}}}} \]

8. Simplified 70.4%

   \[\leadsto \color{blue}{\sqrt{\frac{100}{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{2}}}} \]

9. Taylor expanded in x around 0 5.1%

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

10. Final simplification 5.1%

    \[\leadsto -10 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024100 
(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))))