ENA, Section 1.4, Mentioned, B

Percentage Accurate: 87.8% → 99.6%

Time: 7.2s

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 87.9%

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

   1. sqr-neg 87.9%

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

   2. cancel-sign-sub 87.9%

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

   3. +-commutative 87.9%

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

   4. +-lft-identity 87.9%

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

   5. cancel-sign-sub 87.9%

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

   6. associate-+l- 87.9%

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

   7. sub0-neg 87.9%

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

   8. neg-mul-1 87.9%

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

   9. associate-/r* 87.9%

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

   10. metadata-eval 87.9%

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

   11. sqr-neg 87.9%

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

   12. fma-neg 99.6%

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

   13. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

   \[\leadsto \frac{-10}{\mathsf{fma}\left(x, x, -1\right)} \]
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 87.9%

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

3. Final simplification 87.9%

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

Alternative 3: 13.5% accurate, 2.3× 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 88.3%

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

      1. sqr-neg 88.3%

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

      2. cancel-sign-sub 88.3%

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

      3. +-commutative 88.3%

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

      4. +-lft-identity 88.3%

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

      5. cancel-sign-sub 88.3%

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

      6. associate-+l- 88.3%

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

      7. sub0-neg 88.3%

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

      8. neg-mul-1 88.3%

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

      9. associate-/r* 88.3%

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

      10. metadata-eval 88.3%

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

      11. sqr-neg 88.3%

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

      12. fma-neg 99.6%

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

      13. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 13.5%

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

   if 1 < x

   1. Initial program 87.1%

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

      1. sqr-neg 87.1%

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

      2. cancel-sign-sub 87.1%

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

      3. +-commutative 87.1%

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

      4. +-lft-identity 87.1%

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

      5. cancel-sign-sub 87.1%

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

      6. associate-+l- 87.1%

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

      7. sub0-neg 87.1%

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

      8. neg-mul-1 87.1%

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

      9. associate-/r* 87.1%

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

      10. metadata-eval 87.1%

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

      11. sqr-neg 87.1%

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

      12. fma-neg 99.6%

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

      13. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. frac-2neg 99.6%

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

      2. metadata-eval 99.6%

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

      3. div-inv 99.4%

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

      4. fma-udef 87.1%

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

      5. distribute-neg-in 87.1%

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

      6. metadata-eval 87.1%

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

      7. +-commutative 87.1%

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

      8. sub-neg 87.1%

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

      9. *-commutative 87.1%

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

   6. Applied egg-rr 99.4%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 1.5%

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

      3. frac-times 1.5%

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

      4. metadata-eval 1.5%

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

      5. pow2 1.5%

         \[\leadsto \sqrt{\frac{1}{\color{blue}{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{2}}}} \cdot 10 \]

   8. Applied egg-rr 1.5%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{2}}}} \cdot 10 \]

   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: 5.4% 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. Step-by-step derivation

   1. sqr-neg 87.9%

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

   2. cancel-sign-sub 87.9%

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

   3. +-commutative 87.9%

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

   4. +-lft-identity 87.9%

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

   5. cancel-sign-sub 87.9%

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

   6. associate-+l- 87.9%

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

   7. sub0-neg 87.9%

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

   8. neg-mul-1 87.9%

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

   9. associate-/r* 87.9%

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

   10. metadata-eval 87.9%

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

   11. sqr-neg 87.9%

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

   12. fma-neg 99.6%

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

   13. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. frac-2neg 99.6%

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

   2. metadata-eval 99.6%

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

   3. div-inv 99.4%

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

   4. fma-udef 87.9%

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

   5. distribute-neg-in 87.9%

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

   6. metadata-eval 87.9%

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

   7. +-commutative 87.9%

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

   8. sub-neg 87.9%

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

   9. *-commutative 87.9%

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

6. Applied egg-rr 99.4%

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

   1. add-sqr-sqrt 67.2%

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

   2. sqrt-unprod 68.1%

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

   3. frac-times 68.0%

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

   4. metadata-eval 68.0%

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

   5. pow2 68.0%

      \[\leadsto \sqrt{\frac{1}{\color{blue}{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{2}}}} \cdot 10 \]

8. Applied egg-rr 68.0%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{2}}}} \cdot 10 \]

9. Taylor expanded in x around 0 5.4%

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

10. Final simplification 5.4%

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

Reproduce

herbie shell --seed 2024011 
(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))))