sqrtexp (problem 3.4.4)

Percentage Accurate: 36.4% → 100.0%

Time: 4.9s

Alternatives: 6

Speedup: 21.2×

• could not determine a ground truth

  1. x = 1956684104883356.8

Specification

Math

\[\begin{array}{l} \\ \sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))

C

double code(double x) {
	return sqrt(((exp((2.0 * x)) - 1.0) / (exp(x) - 1.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(((exp((2.0d0 * x)) - 1.0d0) / (exp(x) - 1.0d0)))
end function

Java

public static double code(double x) {
	return Math.sqrt(((Math.exp((2.0 * x)) - 1.0) / (Math.exp(x) - 1.0)));
}

Python

def code(x):
	return math.sqrt(((math.exp((2.0 * x)) - 1.0) / (math.exp(x) - 1.0)))

Julia

function code(x)
	return sqrt(Float64(Float64(exp(Float64(2.0 * x)) - 1.0) / Float64(exp(x) - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt(((exp((2.0 * x)) - 1.0) / (exp(x) - 1.0)));
end

Wolfram

code[x_] := N[Sqrt[N[(N[(N[Exp[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 36.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))

C

double code(double x) {
	return sqrt(((exp((2.0 * x)) - 1.0) / (exp(x) - 1.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(((exp((2.0d0 * x)) - 1.0d0) / (exp(x) - 1.0d0)))
end function

Java

public static double code(double x) {
	return Math.sqrt(((Math.exp((2.0 * x)) - 1.0) / (Math.exp(x) - 1.0)));
}

Python

def code(x):
	return math.sqrt(((math.exp((2.0 * x)) - 1.0) / (math.exp(x) - 1.0)))

Julia

function code(x)
	return sqrt(Float64(Float64(exp(Float64(2.0 * x)) - 1.0) / Float64(exp(x) - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt(((exp((2.0 * x)) - 1.0) / (exp(x) - 1.0)));
end

Wolfram

code[x_] := N[Sqrt[N[(N[(N[Exp[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{{\left(e^{-x}\right)}^{-1} + 1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (+ (pow (exp (- x)) -1.0) 1.0)))

C

double code(double x) {
	return sqrt((pow(exp(-x), -1.0) + 1.0));
}

Fortran

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

Java

public static double code(double x) {
	return Math.sqrt((Math.pow(Math.exp(-x), -1.0) + 1.0));
}

Python

def code(x):
	return math.sqrt((math.pow(math.exp(-x), -1.0) + 1.0))

Julia

function code(x)
	return sqrt(Float64((exp(Float64(-x)) ^ -1.0) + 1.0))
end

MATLAB

function tmp = code(x)
	tmp = sqrt(((exp(-x) ^ -1.0) + 1.0));
end

Wolfram

code[x_] := N[Sqrt[N[(N[Power[N[Exp[(-x)], $MachinePrecision], -1.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 41.1%

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

   1. lift-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}} \]

   2. lift--.f64 N/A

      \[\leadsto \sqrt{\frac{\color{blue}{e^{2 \cdot x} - 1}}{e^{x} - 1}} \]

   3. lift-exp.f64 N/A

      \[\leadsto \sqrt{\frac{\color{blue}{e^{2 \cdot x}} - 1}{e^{x} - 1}} \]

   4. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{e^{\color{blue}{2 \cdot x}} - 1}{e^{x} - 1}} \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{e^{\color{blue}{x \cdot 2}} - 1}{e^{x} - 1}} \]

   6. exp-lft-sqr-rev N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. metadata-eval N/A

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

   10. lift--.f64 N/A

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

   11. flip-+ N/A

       \[\leadsto \sqrt{\color{blue}{e^{x} + 1}} \]

   12. lower-+.f64 100.0

       \[\leadsto \sqrt{\color{blue}{e^{x} + 1}} \]

4. Applied rewrites 100.0%

   \[\leadsto \sqrt{\color{blue}{e^{x} + 1}} \]
5. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \sqrt{\color{blue}{e^{x}} + 1} \]

   2. sinh-+-cosh-rev N/A

      \[\leadsto \sqrt{\color{blue}{\left(\cosh x + \sinh x\right)} + 1} \]

   3. lift-cosh.f64 N/A

      \[\leadsto \sqrt{\left(\color{blue}{\cosh x} + \sinh x\right) + 1} \]

   4. lift-sinh.f64 N/A

      \[\leadsto \sqrt{\left(\cosh x + \color{blue}{\sinh x}\right) + 1} \]

   5. flip-+ N/A

      \[\leadsto \sqrt{\color{blue}{\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{\cosh x - \sinh x}} + 1} \]

   6. lift-cosh.f64 N/A

      \[\leadsto \sqrt{\frac{\color{blue}{\cosh x} \cdot \cosh x - \sinh x \cdot \sinh x}{\cosh x - \sinh x} + 1} \]

   7. lift-cosh.f64 N/A

      \[\leadsto \sqrt{\frac{\cosh x \cdot \color{blue}{\cosh x} - \sinh x \cdot \sinh x}{\cosh x - \sinh x} + 1} \]

   8. lift-sinh.f64 N/A

      \[\leadsto \sqrt{\frac{\cosh x \cdot \cosh x - \color{blue}{\sinh x} \cdot \sinh x}{\cosh x - \sinh x} + 1} \]

   9. lift-sinh.f64 N/A

      \[\leadsto \sqrt{\frac{\cosh x \cdot \cosh x - \sinh x \cdot \color{blue}{\sinh x}}{\cosh x - \sinh x} + 1} \]

   10. sinh-cosh N/A

       \[\leadsto \sqrt{\frac{\color{blue}{1}}{\cosh x - \sinh x} + 1} \]

   11. lift-cosh.f64 N/A

       \[\leadsto \sqrt{\frac{1}{\color{blue}{\cosh x} - \sinh x} + 1} \]

   12. lift-sinh.f64 N/A

       \[\leadsto \sqrt{\frac{1}{\cosh x - \color{blue}{\sinh x}} + 1} \]

   13. sinh---cosh-rev N/A

       \[\leadsto \sqrt{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(x\right)}}} + 1} \]

   14. lower-/.f64 N/A

       \[\leadsto \sqrt{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(x\right)}}} + 1} \]

   15. lower-exp.f64 N/A

       \[\leadsto \sqrt{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(x\right)}}} + 1} \]

   16. lower-neg.f64 100.0

       \[\leadsto \sqrt{\frac{1}{e^{\color{blue}{-x}}} + 1} \]

6. Applied rewrites 100.0%

   \[\leadsto \sqrt{\color{blue}{\frac{1}{e^{-x}}} + 1} \]

7. Final simplification 100.0%

   \[\leadsto \sqrt{{\left(e^{-x}\right)}^{-1} + 1} \]
8. Add Preprocessing

Alternative 2: 98.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x - 1, x, 1\right)\right)}^{-1} + 1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (sqrt
  (+
   (pow (fma (- (* (fma -0.16666666666666666 x 0.5) x) 1.0) x 1.0) -1.0)
   1.0)))

C

double code(double x) {
	return sqrt((pow(fma(((fma(-0.16666666666666666, x, 0.5) * x) - 1.0), x, 1.0), -1.0) + 1.0));
}

Julia

function code(x)
	return sqrt(Float64((fma(Float64(Float64(fma(-0.16666666666666666, x, 0.5) * x) - 1.0), x, 1.0) ^ -1.0) + 1.0))
end

Wolfram

code[x_] := N[Sqrt[N[(N[Power[N[(N[(N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] * x), $MachinePrecision] - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], -1.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 41.1%

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

   1. lift-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}} \]

   2. lift--.f64 N/A

      \[\leadsto \sqrt{\frac{\color{blue}{e^{2 \cdot x} - 1}}{e^{x} - 1}} \]

   3. lift-exp.f64 N/A

      \[\leadsto \sqrt{\frac{\color{blue}{e^{2 \cdot x}} - 1}{e^{x} - 1}} \]

   4. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{e^{\color{blue}{2 \cdot x}} - 1}{e^{x} - 1}} \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{e^{\color{blue}{x \cdot 2}} - 1}{e^{x} - 1}} \]

   6. exp-lft-sqr-rev N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. metadata-eval N/A

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

   10. lift--.f64 N/A

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

   11. flip-+ N/A

       \[\leadsto \sqrt{\color{blue}{e^{x} + 1}} \]

   12. lower-+.f64 100.0

       \[\leadsto \sqrt{\color{blue}{e^{x} + 1}} \]

4. Applied rewrites 100.0%

   \[\leadsto \sqrt{\color{blue}{e^{x} + 1}} \]
5. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \sqrt{\color{blue}{e^{x}} + 1} \]

   2. sinh-+-cosh-rev N/A

      \[\leadsto \sqrt{\color{blue}{\left(\cosh x + \sinh x\right)} + 1} \]

   3. lift-cosh.f64 N/A

      \[\leadsto \sqrt{\left(\color{blue}{\cosh x} + \sinh x\right) + 1} \]

   4. lift-sinh.f64 N/A

      \[\leadsto \sqrt{\left(\cosh x + \color{blue}{\sinh x}\right) + 1} \]

   5. flip-+ N/A

      \[\leadsto \sqrt{\color{blue}{\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{\cosh x - \sinh x}} + 1} \]

   6. lift-cosh.f64 N/A

      \[\leadsto \sqrt{\frac{\color{blue}{\cosh x} \cdot \cosh x - \sinh x \cdot \sinh x}{\cosh x - \sinh x} + 1} \]

   7. lift-cosh.f64 N/A

      \[\leadsto \sqrt{\frac{\cosh x \cdot \color{blue}{\cosh x} - \sinh x \cdot \sinh x}{\cosh x - \sinh x} + 1} \]

   8. lift-sinh.f64 N/A

      \[\leadsto \sqrt{\frac{\cosh x \cdot \cosh x - \color{blue}{\sinh x} \cdot \sinh x}{\cosh x - \sinh x} + 1} \]

   9. lift-sinh.f64 N/A

      \[\leadsto \sqrt{\frac{\cosh x \cdot \cosh x - \sinh x \cdot \color{blue}{\sinh x}}{\cosh x - \sinh x} + 1} \]

   10. sinh-cosh N/A

       \[\leadsto \sqrt{\frac{\color{blue}{1}}{\cosh x - \sinh x} + 1} \]

   11. lift-cosh.f64 N/A

       \[\leadsto \sqrt{\frac{1}{\color{blue}{\cosh x} - \sinh x} + 1} \]

   12. lift-sinh.f64 N/A

       \[\leadsto \sqrt{\frac{1}{\cosh x - \color{blue}{\sinh x}} + 1} \]

   13. sinh---cosh-rev N/A

       \[\leadsto \sqrt{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(x\right)}}} + 1} \]

   14. lower-/.f64 N/A

       \[\leadsto \sqrt{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(x\right)}}} + 1} \]

   15. lower-exp.f64 N/A

       \[\leadsto \sqrt{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(x\right)}}} + 1} \]

   16. lower-neg.f64 100.0

       \[\leadsto \sqrt{\frac{1}{e^{\color{blue}{-x}}} + 1} \]

6. Applied rewrites 100.0%

   \[\leadsto \sqrt{\color{blue}{\frac{1}{e^{-x}}} + 1} \]

7. Taylor expanded in x around 0

   \[\leadsto \sqrt{\frac{1}{\color{blue}{1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right)}} + 1} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right) + 1}} + 1} \]

   2. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right) \cdot x} + 1} + 1} \]

   3. lower-fma.f64 N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1, x, 1\right)}} + 1} \]

   4. lower--.f64 N/A

      \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1}, x, 1\right)} + 1} \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) \cdot x} - 1, x, 1\right)} + 1} \]

   6. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) \cdot x} - 1, x, 1\right)} + 1} \]

   7. +-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{6} \cdot x + \frac{1}{2}\right)} \cdot x - 1, x, 1\right)} + 1} \]

   8. lower-fma.f64 97.9

      \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right)} \cdot x - 1, x, 1\right)} + 1} \]

9. Applied rewrites 97.9%

   \[\leadsto \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x - 1, x, 1\right)}} + 1} \]

10. Final simplification 97.9%

    \[\leadsto \sqrt{{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x - 1, x, 1\right)\right)}^{-1} + 1} \]
11. Add Preprocessing

Alternative 3: 98.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{{\left(\mathsf{fma}\left(0.5 \cdot x - 1, x, 1\right)\right)}^{-1} + 1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (sqrt (+ (pow (fma (- (* 0.5 x) 1.0) x 1.0) -1.0) 1.0)))

C

double code(double x) {
	return sqrt((pow(fma(((0.5 * x) - 1.0), x, 1.0), -1.0) + 1.0));
}

Julia

function code(x)
	return sqrt(Float64((fma(Float64(Float64(0.5 * x) - 1.0), x, 1.0) ^ -1.0) + 1.0))
end

Wolfram

code[x_] := N[Sqrt[N[(N[Power[N[(N[(N[(0.5 * x), $MachinePrecision] - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], -1.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 41.1%

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

   1. lift-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}} \]

   2. lift--.f64 N/A

      \[\leadsto \sqrt{\frac{\color{blue}{e^{2 \cdot x} - 1}}{e^{x} - 1}} \]

   3. lift-exp.f64 N/A

      \[\leadsto \sqrt{\frac{\color{blue}{e^{2 \cdot x}} - 1}{e^{x} - 1}} \]

   4. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{e^{\color{blue}{2 \cdot x}} - 1}{e^{x} - 1}} \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{e^{\color{blue}{x \cdot 2}} - 1}{e^{x} - 1}} \]

   6. exp-lft-sqr-rev N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. metadata-eval N/A

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

   10. lift--.f64 N/A

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

   11. flip-+ N/A

       \[\leadsto \sqrt{\color{blue}{e^{x} + 1}} \]

   12. lower-+.f64 100.0

       \[\leadsto \sqrt{\color{blue}{e^{x} + 1}} \]

4. Applied rewrites 100.0%

   \[\leadsto \sqrt{\color{blue}{e^{x} + 1}} \]
5. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \sqrt{\color{blue}{e^{x}} + 1} \]

   2. sinh-+-cosh-rev N/A

      \[\leadsto \sqrt{\color{blue}{\left(\cosh x + \sinh x\right)} + 1} \]

   3. lift-cosh.f64 N/A

      \[\leadsto \sqrt{\left(\color{blue}{\cosh x} + \sinh x\right) + 1} \]

   4. lift-sinh.f64 N/A

      \[\leadsto \sqrt{\left(\cosh x + \color{blue}{\sinh x}\right) + 1} \]

   5. flip-+ N/A

      \[\leadsto \sqrt{\color{blue}{\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{\cosh x - \sinh x}} + 1} \]

   6. lift-cosh.f64 N/A

      \[\leadsto \sqrt{\frac{\color{blue}{\cosh x} \cdot \cosh x - \sinh x \cdot \sinh x}{\cosh x - \sinh x} + 1} \]

   7. lift-cosh.f64 N/A

      \[\leadsto \sqrt{\frac{\cosh x \cdot \color{blue}{\cosh x} - \sinh x \cdot \sinh x}{\cosh x - \sinh x} + 1} \]

   8. lift-sinh.f64 N/A

      \[\leadsto \sqrt{\frac{\cosh x \cdot \cosh x - \color{blue}{\sinh x} \cdot \sinh x}{\cosh x - \sinh x} + 1} \]

   9. lift-sinh.f64 N/A

      \[\leadsto \sqrt{\frac{\cosh x \cdot \cosh x - \sinh x \cdot \color{blue}{\sinh x}}{\cosh x - \sinh x} + 1} \]

   10. sinh-cosh N/A

       \[\leadsto \sqrt{\frac{\color{blue}{1}}{\cosh x - \sinh x} + 1} \]

   11. lift-cosh.f64 N/A

       \[\leadsto \sqrt{\frac{1}{\color{blue}{\cosh x} - \sinh x} + 1} \]

   12. lift-sinh.f64 N/A

       \[\leadsto \sqrt{\frac{1}{\cosh x - \color{blue}{\sinh x}} + 1} \]

   13. sinh---cosh-rev N/A

       \[\leadsto \sqrt{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(x\right)}}} + 1} \]

   14. lower-/.f64 N/A

       \[\leadsto \sqrt{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(x\right)}}} + 1} \]

   15. lower-exp.f64 N/A

       \[\leadsto \sqrt{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(x\right)}}} + 1} \]

   16. lower-neg.f64 100.0

       \[\leadsto \sqrt{\frac{1}{e^{\color{blue}{-x}}} + 1} \]

6. Applied rewrites 100.0%

   \[\leadsto \sqrt{\color{blue}{\frac{1}{e^{-x}}} + 1} \]

7. Taylor expanded in x around 0

   \[\leadsto \sqrt{\frac{1}{\color{blue}{1 + x \cdot \left(\frac{1}{2} \cdot x - 1\right)}} + 1} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{x \cdot \left(\frac{1}{2} \cdot x - 1\right) + 1}} + 1} \]

   2. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\left(\frac{1}{2} \cdot x - 1\right) \cdot x} + 1} + 1} \]

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 97.5

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

9. Applied rewrites 97.5%

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

10. Final simplification 97.5%

    \[\leadsto \sqrt{{\left(\mathsf{fma}\left(0.5 \cdot x - 1, x, 1\right)\right)}^{-1} + 1} \]
11. Add Preprocessing

Alternative 4: 97.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{{\left(1 - x\right)}^{-1} + 1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (+ (pow (- 1.0 x) -1.0) 1.0)))

C

double code(double x) {
	return sqrt((pow((1.0 - x), -1.0) + 1.0));
}

Fortran

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

Java

public static double code(double x) {
	return Math.sqrt((Math.pow((1.0 - x), -1.0) + 1.0));
}

Python

def code(x):
	return math.sqrt((math.pow((1.0 - x), -1.0) + 1.0))

Julia

function code(x)
	return sqrt(Float64((Float64(1.0 - x) ^ -1.0) + 1.0))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((((1.0 - x) ^ -1.0) + 1.0));
end

Wolfram

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

Derivation

1. Initial program 41.1%

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

   1. lift-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}} \]

   2. lift--.f64 N/A

      \[\leadsto \sqrt{\frac{\color{blue}{e^{2 \cdot x} - 1}}{e^{x} - 1}} \]

   3. lift-exp.f64 N/A

      \[\leadsto \sqrt{\frac{\color{blue}{e^{2 \cdot x}} - 1}{e^{x} - 1}} \]

   4. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{e^{\color{blue}{2 \cdot x}} - 1}{e^{x} - 1}} \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{e^{\color{blue}{x \cdot 2}} - 1}{e^{x} - 1}} \]

   6. exp-lft-sqr-rev N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. metadata-eval N/A

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

   10. lift--.f64 N/A

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

   11. flip-+ N/A

       \[\leadsto \sqrt{\color{blue}{e^{x} + 1}} \]

   12. lower-+.f64 100.0

       \[\leadsto \sqrt{\color{blue}{e^{x} + 1}} \]

4. Applied rewrites 100.0%

   \[\leadsto \sqrt{\color{blue}{e^{x} + 1}} \]
5. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \sqrt{\color{blue}{e^{x}} + 1} \]

   2. sinh-+-cosh-rev N/A

      \[\leadsto \sqrt{\color{blue}{\left(\cosh x + \sinh x\right)} + 1} \]

   3. lift-cosh.f64 N/A

      \[\leadsto \sqrt{\left(\color{blue}{\cosh x} + \sinh x\right) + 1} \]

   4. lift-sinh.f64 N/A

      \[\leadsto \sqrt{\left(\cosh x + \color{blue}{\sinh x}\right) + 1} \]

   5. flip-+ N/A

      \[\leadsto \sqrt{\color{blue}{\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{\cosh x - \sinh x}} + 1} \]

   6. lift-cosh.f64 N/A

      \[\leadsto \sqrt{\frac{\color{blue}{\cosh x} \cdot \cosh x - \sinh x \cdot \sinh x}{\cosh x - \sinh x} + 1} \]

   7. lift-cosh.f64 N/A

      \[\leadsto \sqrt{\frac{\cosh x \cdot \color{blue}{\cosh x} - \sinh x \cdot \sinh x}{\cosh x - \sinh x} + 1} \]

   8. lift-sinh.f64 N/A

      \[\leadsto \sqrt{\frac{\cosh x \cdot \cosh x - \color{blue}{\sinh x} \cdot \sinh x}{\cosh x - \sinh x} + 1} \]

   9. lift-sinh.f64 N/A

      \[\leadsto \sqrt{\frac{\cosh x \cdot \cosh x - \sinh x \cdot \color{blue}{\sinh x}}{\cosh x - \sinh x} + 1} \]

   10. sinh-cosh N/A

       \[\leadsto \sqrt{\frac{\color{blue}{1}}{\cosh x - \sinh x} + 1} \]

   11. lift-cosh.f64 N/A

       \[\leadsto \sqrt{\frac{1}{\color{blue}{\cosh x} - \sinh x} + 1} \]

   12. lift-sinh.f64 N/A

       \[\leadsto \sqrt{\frac{1}{\cosh x - \color{blue}{\sinh x}} + 1} \]

   13. sinh---cosh-rev N/A

       \[\leadsto \sqrt{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(x\right)}}} + 1} \]

   14. lower-/.f64 N/A

       \[\leadsto \sqrt{\color{blue}{\frac{1}{e^{\mathsf{neg}\left(x\right)}}} + 1} \]

   15. lower-exp.f64 N/A

       \[\leadsto \sqrt{\frac{1}{\color{blue}{e^{\mathsf{neg}\left(x\right)}}} + 1} \]

   16. lower-neg.f64 100.0

       \[\leadsto \sqrt{\frac{1}{e^{\color{blue}{-x}}} + 1} \]

6. Applied rewrites 100.0%

   \[\leadsto \sqrt{\color{blue}{\frac{1}{e^{-x}}} + 1} \]

7. Taylor expanded in x around 0

   \[\leadsto \sqrt{\frac{1}{\color{blue}{1 + -1 \cdot x}} + 1} \]
8. Step-by-step derivation

   1. *-commutative N/A

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

   2. fp-cancel-sign-sub-inv N/A

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

   3. *-commutative N/A

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

   4. mul-1-neg N/A

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

   5. remove-double-neg N/A

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

   6. lower--.f64 96.7

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

9. Applied rewrites 96.7%

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

10. Final simplification 96.7%

    \[\leadsto \sqrt{{\left(1 - x\right)}^{-1} + 1} \]
11. Add Preprocessing

Alternative 5: 100.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{e^{x} + 1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (+ (exp x) 1.0)))

C

double code(double x) {
	return sqrt((exp(x) + 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((exp(x) + 1.0d0))
end function

Java

public static double code(double x) {
	return Math.sqrt((Math.exp(x) + 1.0));
}

Python

def code(x):
	return math.sqrt((math.exp(x) + 1.0))

Julia

function code(x)
	return sqrt(Float64(exp(x) + 1.0))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((exp(x) + 1.0));
end

Wolfram

code[x_] := N[Sqrt[N[(N[Exp[x], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 41.1%

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

   1. lift-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}} \]

   2. lift--.f64 N/A

      \[\leadsto \sqrt{\frac{\color{blue}{e^{2 \cdot x} - 1}}{e^{x} - 1}} \]

   3. lift-exp.f64 N/A

      \[\leadsto \sqrt{\frac{\color{blue}{e^{2 \cdot x}} - 1}{e^{x} - 1}} \]

   4. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{e^{\color{blue}{2 \cdot x}} - 1}{e^{x} - 1}} \]

   5. *-commutative N/A

      \[\leadsto \sqrt{\frac{e^{\color{blue}{x \cdot 2}} - 1}{e^{x} - 1}} \]

   6. exp-lft-sqr-rev N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. metadata-eval N/A

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

   10. lift--.f64 N/A

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

   11. flip-+ N/A

       \[\leadsto \sqrt{\color{blue}{e^{x} + 1}} \]

   12. lower-+.f64 100.0

       \[\leadsto \sqrt{\color{blue}{e^{x} + 1}} \]

4. Applied rewrites 100.0%

   \[\leadsto \sqrt{\color{blue}{e^{x} + 1}} \]
5. Add Preprocessing

Alternative 6: 71.2% accurate, 21.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt 2.0))

C

double code(double x) {
	return sqrt(2.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(2.0d0)
end function

Java

public static double code(double x) {
	return Math.sqrt(2.0);
}

Python

def code(x):
	return math.sqrt(2.0)

Julia

function code(x)
	return sqrt(2.0)
end

MATLAB

function tmp = code(x)
	tmp = sqrt(2.0);
end

Wolfram

code[x_] := N[Sqrt[2.0], $MachinePrecision]

Derivation

1. Initial program 41.1%

   \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \sqrt{\color{blue}{2}} \]
4. Step-by-step derivation

5. Applied rewrites 68.1%

   \[\leadsto \sqrt{\color{blue}{2}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024342 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))