ENA, Section 1.4, Mentioned, B

Percentage Accurate: 87.7% → 99.6%

Time: 4.4s

Alternatives: 5

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: 87.7% 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 87.6%

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

   1. sub-neg 87.6%

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

   2. +-commutative 87.6%

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

   3. neg-sub0 87.6%

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

   4. associate-+l- 87.6%

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

   5. sub0-neg 87.6%

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

   6. neg-mul-1 87.6%

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

   7. associate-/r* 87.6%

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

   8. metadata-eval 87.6%

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

   9. fma-neg 99.6%

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

   10. 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. Final simplification 99.6%

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

Alternative 2: 13.5% accurate, 0.8× 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 88.0%

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

      1. sub-neg 88.0%

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

      2. +-commutative 88.0%

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

      3. neg-sub0 88.0%

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

      4. associate-+l- 88.0%

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

      5. sub0-neg 88.0%

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

      6. neg-mul-1 88.0%

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

      7. associate-/r* 88.0%

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

      8. metadata-eval 88.0%

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

      9. fma-neg 99.6%

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

      10. 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. Taylor expanded in x around 0 13.4%

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

   if 1 < (*.f64 x x)

   1. Initial program 86.8%

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

      1. sub-neg 86.8%

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

      2. +-commutative 86.8%

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

      3. neg-sub0 86.8%

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

      4. associate-+l- 86.8%

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

      5. sub0-neg 86.8%

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

      6. neg-mul-1 86.8%

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

      7. associate-/r* 86.8%

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

      8. metadata-eval 86.8%

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

      9. fma-neg 99.5%

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

      10. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 13.5%

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

      1. unpow2 13.5%

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

   6. Simplified 13.5%

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

4. Final simplification 13.5%

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

Alternative 3: 87.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.6%

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

   1. sub-neg 87.6%

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

   2. +-commutative 87.6%

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

   3. neg-sub0 87.6%

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

   4. associate-+l- 87.6%

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

   5. sub0-neg 87.6%

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

   6. neg-mul-1 87.6%

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

   7. associate-/r* 87.6%

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

   8. metadata-eval 87.6%

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

   9. fma-neg 99.6%

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

   10. 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. 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. fma-udef 87.6%

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

   4. distribute-neg-in 87.6%

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

   5. metadata-eval 87.6%

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

   6. +-commutative 87.6%

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

   7. sub-neg 87.6%

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

   8. div-inv 87.6%

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

   9. *-commutative 87.6%

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

5. Applied egg-rr 87.6%

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

6. Final simplification 87.6%

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

Alternative 4: 87.7% 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 87.6%

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

2. Final simplification 87.6%

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

Alternative 5: 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 87.6%

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

   1. sub-neg 87.6%

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

   2. +-commutative 87.6%

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

   3. neg-sub0 87.6%

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

   4. associate-+l- 87.6%

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

   5. sub0-neg 87.6%

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

   6. neg-mul-1 87.6%

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

   7. associate-/r* 87.6%

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

   8. metadata-eval 87.6%

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

   9. fma-neg 99.6%

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

   10. 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. Taylor expanded in x around 0 9.3%

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

5. Final simplification 9.3%

   \[\leadsto 10 \]

Reproduce

herbie shell --seed 2023178 
(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))))