sqrtexp (problem 3.4.4)

Percentage Accurate: 35.7% → 100.0%

Time: 7.9s

Alternatives: 5

Speedup: 21.2×

• could not determine a ground truth

  1. x = 1.1215272163715295e+30

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: 35.7% 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, 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 37.6%

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

   1. *-commutative N/A

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

   2. exp-lft-sqr N/A

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

   3. metadata-eval N/A

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

   4. flip-+ N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. --lowering--.f64 N/A

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

   9. exp-lowering-exp.f64 N/A

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

   10. metadata-eval 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 72.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-1 + e^{x \cdot 2}}{e^{x} + -1} \leq 1:\\ \;\;\;\;\sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\sqrt{2}}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.036458333333333336, 0.1875\right), 0.5\right), \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ -1.0 (exp (* x 2.0))) (+ (exp x) -1.0)) 1.0)
   (sqrt 2.0)
   (fma
    (/ x (sqrt 2.0))
    (fma x (fma x 0.036458333333333336 0.1875) 0.5)
    (sqrt 2.0))))

C

double code(double x) {
	double tmp;
	if (((-1.0 + exp((x * 2.0))) / (exp(x) + -1.0)) <= 1.0) {
		tmp = sqrt(2.0);
	} else {
		tmp = fma((x / sqrt(2.0)), fma(x, fma(x, 0.036458333333333336, 0.1875), 0.5), sqrt(2.0));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(-1.0 + exp(Float64(x * 2.0))) / Float64(exp(x) + -1.0)) <= 1.0)
		tmp = sqrt(2.0);
	else
		tmp = fma(Float64(x / sqrt(2.0)), fma(x, fma(x, 0.036458333333333336, 0.1875), 0.5), sqrt(2.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(-1.0 + N[Exp[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], 1.0], N[Sqrt[2.0], $MachinePrecision], N[(N[(x / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * 0.036458333333333336 + 0.1875), $MachinePrecision] + 0.5), $MachinePrecision] + N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (exp.f64 (*.f64 #s(literal 2 binary64) x)) #s(literal 1 binary64)) (-.f64 (exp.f64 x) #s(literal 1 binary64))) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 20.7%

      \[\leadsto \sqrt{\color{blue}{2}} \]

   if 1 < (/.f64 (-.f64 (exp.f64 (*.f64 #s(literal 2 binary64) x)) #s(literal 1 binary64)) (-.f64 (exp.f64 x) #s(literal 1 binary64)))

   1. Initial program 9.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\sqrt{2} + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{x \cdot \left(\frac{1}{6} - \frac{1}{2} \cdot \frac{\frac{1}{2} - \frac{1}{4} \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}}{{\left(\sqrt{2}\right)}^{2}}\right)}{\sqrt{2}} + \frac{1}{2} \cdot \frac{\frac{1}{2} - \frac{1}{4} \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}}{\sqrt{2}}\right) + \frac{1}{2} \cdot \frac{1}{\sqrt{2}}\right)} \]

   5. Simplified 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{2}}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.036458333333333336, 0.1875\right), 0.5\right), \sqrt{2}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-1 + e^{x \cdot 2}}{e^{x} + -1} \leq 1:\\ \;\;\;\;\sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\sqrt{2}}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.036458333333333336, 0.1875\right), 0.5\right), \sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-1 + e^{x \cdot 2}}{e^{x} + -1} \leq 1.2:\\ \;\;\;\;\sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), x \cdot x, x + 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ -1.0 (exp (* x 2.0))) (+ (exp x) -1.0)) 1.2)
   (sqrt 2.0)
   (sqrt (fma (fma x 0.16666666666666666 0.5) (* x x) (+ x 2.0)))))

C

double code(double x) {
	double tmp;
	if (((-1.0 + exp((x * 2.0))) / (exp(x) + -1.0)) <= 1.2) {
		tmp = sqrt(2.0);
	} else {
		tmp = sqrt(fma(fma(x, 0.16666666666666666, 0.5), (x * x), (x + 2.0)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(-1.0 + exp(Float64(x * 2.0))) / Float64(exp(x) + -1.0)) <= 1.2)
		tmp = sqrt(2.0);
	else
		tmp = sqrt(fma(fma(x, 0.16666666666666666, 0.5), Float64(x * x), Float64(x + 2.0)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(-1.0 + N[Exp[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], 1.2], N[Sqrt[2.0], $MachinePrecision], N[Sqrt[N[(N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (exp.f64 (*.f64 #s(literal 2 binary64) x)) #s(literal 1 binary64)) (-.f64 (exp.f64 x) #s(literal 1 binary64))) < 1.19999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 20.7%

      \[\leadsto \sqrt{\color{blue}{2}} \]

   if 1.19999999999999996 < (/.f64 (-.f64 (exp.f64 (*.f64 #s(literal 2 binary64) x)) #s(literal 1 binary64)) (-.f64 (exp.f64 x) #s(literal 1 binary64)))

   1. Initial program 8.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 97.3

         \[\leadsto \sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 2\right)} \]

   5. Simplified 97.3%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 2\right)}} \]
   6. Step-by-step derivation

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 97.3

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), x \cdot x, \color{blue}{x + 2}\right)} \]

   7. Applied egg-rr 97.3%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), x \cdot x, x + 2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-1 + e^{x \cdot 2}}{e^{x} + -1} \leq 1.2:\\ \;\;\;\;\sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), x \cdot x, x + 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-1 + e^{x \cdot 2}}{e^{x} + -1} \leq 1.2:\\ \;\;\;\;\sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (+ -1.0 (exp (* x 2.0))) (+ (exp x) -1.0)) 1.2)
   (sqrt 2.0)
   (sqrt (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 2.0))))

C

double code(double x) {
	double tmp;
	if (((-1.0 + exp((x * 2.0))) / (exp(x) + -1.0)) <= 1.2) {
		tmp = sqrt(2.0);
	} else {
		tmp = sqrt(fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 2.0));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(-1.0 + exp(Float64(x * 2.0))) / Float64(exp(x) + -1.0)) <= 1.2)
		tmp = sqrt(2.0);
	else
		tmp = sqrt(fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 2.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(-1.0 + N[Exp[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], 1.2], N[Sqrt[2.0], $MachinePrecision], N[Sqrt[N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (exp.f64 (*.f64 #s(literal 2 binary64) x)) #s(literal 1 binary64)) (-.f64 (exp.f64 x) #s(literal 1 binary64))) < 1.19999999999999996

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 20.7%

      \[\leadsto \sqrt{\color{blue}{2}} \]

   if 1.19999999999999996 < (/.f64 (-.f64 (exp.f64 (*.f64 #s(literal 2 binary64) x)) #s(literal 1 binary64)) (-.f64 (exp.f64 x) #s(literal 1 binary64)))

   1. Initial program 8.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 97.3

         \[\leadsto \sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right), 2\right)} \]

   5. Simplified 97.3%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-1 + e^{x \cdot 2}}{e^{x} + -1} \leq 1.2:\\ \;\;\;\;\sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.7% 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 37.6%

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

3. Taylor expanded in x around 0

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

5. Simplified 71.4%

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

Reproduce

herbie shell --seed 2024205 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))