ENA, Section 1.4, Mentioned, B

Percentage Accurate: 87.7% → 99.6%

Time: 9.0s

Alternatives: 8

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 88.0%

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

   1. sqr-neg 88.0%

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

   2. remove-double-neg 88.0%

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

   3. distribute-neg-frac2 88.0%

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

   4. distribute-neg-frac 88.0%

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

   5. metadata-eval 88.0%

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

   6. neg-sub0 88.0%

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

   7. associate--r- 88.0%

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

   8. metadata-eval 88.0%

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

   9. +-commutative 88.0%

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

   10. sqr-neg 88.0%

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

   11. fma-define 99.6%

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

3. Simplified 99.6%

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

Alternative 2: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

   1. sqr-neg 88.0%

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

   2. remove-double-neg 88.0%

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

   3. distribute-neg-frac2 88.0%

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

   4. distribute-neg-frac 88.0%

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

   5. metadata-eval 88.0%

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

   6. neg-sub0 88.0%

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

   7. associate--r- 88.0%

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

   8. metadata-eval 88.0%

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

   9. +-commutative 88.0%

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

   10. sqr-neg 88.0%

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

   11. fma-define 99.6%

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

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. fma-undefine 88.0%

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

   2. difference-of-sqr--1 99.5%

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

6. Applied egg-rr 99.5%

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

   1. *-commutative 99.5%

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

   2. +-commutative 99.5%

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

   3. distribute-lft-in 99.5%

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

   4. *-commutative 99.5%

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

   5. *-un-lft-identity 99.5%

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

   6. sub-neg 99.5%

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

   7. metadata-eval 99.5%

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

   8. sub-neg 99.5%

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

   9. metadata-eval 99.5%

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

8. Applied egg-rr 99.5%

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

9. Final simplification 99.5%

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

Alternative 3: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

   1. sqr-neg 88.0%

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

   2. remove-double-neg 88.0%

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

   3. distribute-neg-frac2 88.0%

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

   4. distribute-neg-frac 88.0%

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

   5. metadata-eval 88.0%

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

   6. neg-sub0 88.0%

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

   7. associate--r- 88.0%

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

   8. metadata-eval 88.0%

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

   9. +-commutative 88.0%

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

   10. sqr-neg 88.0%

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

   11. fma-define 99.6%

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

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. fma-undefine 88.0%

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

   4. distribute-neg-in 88.0%

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

   5. metadata-eval 88.0%

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

   6. +-commutative 88.0%

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

   7. sub-neg 88.0%

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

   8. add-sqr-sqrt 58.8%

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

   9. sqrt-unprod 59.3%

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

   10. pow1/2 59.3%

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

6. Applied egg-rr 66.6%

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

   1. unpow1/2 66.6%

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

8. Simplified 66.6%

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

   1. metadata-eval 66.6%

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

   2. unpow2 66.6%

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

   3. frac-times 66.7%

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

   4. sqrt-unprod 65.7%

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

   5. add-sqr-sqrt 99.6%

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

   6. fma-undefine 88.0%

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

   7. difference-of-sqr--1 99.5%

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

   8. *-commutative 99.5%

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

   9. associate-/r* 99.5%

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

   10. sub-neg 99.5%

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

   11. metadata-eval 99.5%

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

10. Applied egg-rr 99.5%

    \[\leadsto \color{blue}{\frac{\frac{-10}{x + -1}}{x + 1}} \]
11. Add Preprocessing

Alternative 4: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

   1. sqr-neg 88.0%

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

   2. remove-double-neg 88.0%

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

   3. distribute-neg-frac2 88.0%

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

   4. distribute-neg-frac 88.0%

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

   5. metadata-eval 88.0%

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

   6. neg-sub0 88.0%

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

   7. associate--r- 88.0%

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

   8. metadata-eval 88.0%

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

   9. +-commutative 88.0%

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

   10. sqr-neg 88.0%

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

   11. fma-define 99.6%

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

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. fma-undefine 88.0%

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

   2. difference-of-sqr--1 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 5: 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 88.0%

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

Alternative 6: 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 88.6%

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

      1. sqr-neg 88.6%

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

      2. remove-double-neg 88.6%

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

      3. distribute-neg-frac2 88.6%

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

      4. distribute-neg-frac 88.6%

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

      5. metadata-eval 88.6%

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

      6. neg-sub0 88.6%

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

      7. associate--r- 88.6%

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

      8. metadata-eval 88.6%

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

      9. +-commutative 88.6%

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

      10. sqr-neg 88.6%

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

      11. fma-define 99.6%

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

   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 86.8%

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

      1. sqr-neg 86.8%

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

      2. remove-double-neg 86.8%

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

      3. distribute-neg-frac2 86.8%

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

      4. distribute-neg-frac 86.8%

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

      5. metadata-eval 86.8%

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

      6. neg-sub0 86.8%

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

      7. associate--r- 86.8%

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

      8. metadata-eval 86.8%

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

      9. +-commutative 86.8%

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

      10. sqr-neg 86.8%

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

      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 86.8%

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

      4. distribute-neg-in 86.8%

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

      5. metadata-eval 86.8%

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

      6. +-commutative 86.8%

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

      7. sub-neg 86.8%

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

      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. Add Preprocessing

Alternative 7: 18.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

   1. sqr-neg 88.0%

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

   2. remove-double-neg 88.0%

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

   3. distribute-neg-frac2 88.0%

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

   4. distribute-neg-frac 88.0%

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

   5. metadata-eval 88.0%

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

   6. neg-sub0 88.0%

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

   7. associate--r- 88.0%

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

   8. metadata-eval 88.0%

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

   9. +-commutative 88.0%

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

   10. sqr-neg 88.0%

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

   11. fma-define 99.6%

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

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. fma-undefine 88.0%

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

   4. distribute-neg-in 88.0%

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

   5. metadata-eval 88.0%

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

   6. +-commutative 88.0%

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

   7. sub-neg 88.0%

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

   8. add-sqr-sqrt 58.8%

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

   9. sqrt-unprod 59.3%

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

   10. pow1/2 59.3%

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

6. Applied egg-rr 66.6%

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

   1. unpow1/2 66.6%

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

8. Simplified 66.6%

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

   1. metadata-eval 66.6%

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

   2. unpow2 66.6%

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

   3. frac-times 66.7%

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

   4. sqrt-unprod 65.7%

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

   5. add-sqr-sqrt 99.6%

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

   6. fma-undefine 88.0%

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

   7. difference-of-sqr--1 99.5%

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

   8. associate-/r* 99.5%

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

   9. sub-neg 99.5%

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

   10. metadata-eval 99.5%

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

10. Applied egg-rr 99.5%

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

11. Taylor expanded in x around 0 18.8%

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

Alternative 8: 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 88.0%

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

   1. sqr-neg 88.0%

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

   2. remove-double-neg 88.0%

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

   3. distribute-neg-frac2 88.0%

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

   4. distribute-neg-frac 88.0%

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

   5. metadata-eval 88.0%

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

   6. neg-sub0 88.0%

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

   7. associate--r- 88.0%

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

   8. metadata-eval 88.0%

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

   9. +-commutative 88.0%

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

   10. sqr-neg 88.0%

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

   11. fma-define 99.6%

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

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. fma-undefine 88.0%

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

   4. distribute-neg-in 88.0%

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

   5. metadata-eval 88.0%

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

   6. +-commutative 88.0%

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

   7. sub-neg 88.0%

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

   8. add-sqr-sqrt 58.8%

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

   9. sqrt-unprod 59.3%

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

   10. pow1/2 59.3%

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

6. Applied egg-rr 66.6%

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

   1. unpow1/2 66.6%

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

8. Simplified 66.6%

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

9. Taylor expanded in x around 0 5.6%

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

Reproduce

herbie shell --seed 2024182 
(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))))