sqrtexp (problem 3.4.4)

Percentage Accurate: 35.6% → 100.0%

Time: 4.6s

Alternatives: 9

Speedup: 43.8×

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: 35.6% 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, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{hypot}\left(1, e^{x \cdot 0.5}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (hypot 1.0 (exp (* x 0.5))))

C

double code(double x) {
	return hypot(1.0, exp((x * 0.5)));
}

Java

public static double code(double x) {
	return Math.hypot(1.0, Math.exp((x * 0.5)));
}

Python

def code(x):
	return math.hypot(1.0, math.exp((x * 0.5)))

Julia

function code(x)
	return hypot(1.0, exp(Float64(x * 0.5)))
end

MATLAB

function tmp = code(x)
	tmp = hypot(1.0, exp((x * 0.5)));
end

Wolfram

code[x_] := N[Sqrt[1.0 ^ 2 + N[Exp[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]

Derivation

1. Initial program 30.7%

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

   1. *-commutative 30.7%

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

   2. exp-lft-sqr 30.7%

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

   3. difference-of-sqr-1 30.9%

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

   4. associate-/l* 31.2%

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

   5. *-inverses 99.6%

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

   6. /-rgt-identity 99.6%

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

   7. +-commutative 99.6%

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

3. Simplified 99.6%

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

   1. +-commutative 99.6%

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

   2. flip-+ 30.7%

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

   3. metadata-eval 30.7%

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

   4. prod-exp 30.7%

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

   5. expm1-udef 31.6%

      \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{expm1}\left(x + x\right)}}{e^{x} - 1}} \]

   6. expm1-udef 98.8%

      \[\leadsto \sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\color{blue}{\mathsf{expm1}\left(x\right)}}} \]

   7. div-inv 98.7%

      \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(x + x\right) \cdot \frac{1}{\mathsf{expm1}\left(x\right)}}} \]

   8. div-inv 98.8%

      \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

   9. add-log-exp 98.8%

      \[\leadsto \color{blue}{\log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

   10. *-un-lft-identity 98.8%

       \[\leadsto \log \color{blue}{\left(1 \cdot e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

   11. log-prod 98.8%

       \[\leadsto \color{blue}{\log 1 + \log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

   12. metadata-eval 98.8%

       \[\leadsto \color{blue}{0} + \log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right) \]

   13. add-log-exp 98.8%

       \[\leadsto 0 + \color{blue}{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

   14. div-inv 98.7%

       \[\leadsto 0 + \sqrt{\color{blue}{\mathsf{expm1}\left(x + x\right) \cdot \frac{1}{\mathsf{expm1}\left(x\right)}}} \]

   15. div-inv 98.8%

       \[\leadsto 0 + \sqrt{\color{blue}{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

   16. expm1-udef 32.6%

       \[\leadsto 0 + \sqrt{\frac{\color{blue}{e^{x + x} - 1}}{\mathsf{expm1}\left(x\right)}} \]

   17. prod-exp 32.6%

       \[\leadsto 0 + \sqrt{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{\mathsf{expm1}\left(x\right)}} \]

   18. metadata-eval 32.6%

       \[\leadsto 0 + \sqrt{\frac{e^{x} \cdot e^{x} - \color{blue}{1 \cdot 1}}{\mathsf{expm1}\left(x\right)}} \]

5. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{0 + \sqrt{1 + e^{x}}} \]
6. Step-by-step derivation

   1. +-lft-identity 99.6%

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

   2. rem-square-sqrt 99.6%

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

   3. hypot-1-def 99.6%

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

7. Simplified 99.6%

   \[\leadsto \color{blue}{\mathsf{hypot}\left(1, \sqrt{e^{x}}\right)} \]
8. Step-by-step derivation

   1. pow1/2 99.6%

      \[\leadsto \mathsf{hypot}\left(1, \color{blue}{{\left(e^{x}\right)}^{0.5}}\right) \]

   2. pow-exp 100.0%

      \[\leadsto \mathsf{hypot}\left(1, \color{blue}{e^{x \cdot 0.5}}\right) \]

9. Applied egg-rr 100.0%

   \[\leadsto \mathsf{hypot}\left(1, \color{blue}{e^{x \cdot 0.5}}\right) \]

10. Final simplification 100.0%

    \[\leadsto \mathsf{hypot}\left(1, e^{x \cdot 0.5}\right) \]

Alternative 2: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.7%

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

   1. *-commutative 30.7%

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

   2. exp-lft-sqr 30.7%

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

   3. difference-of-sqr-1 30.9%

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

   4. associate-/l* 31.2%

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

   5. *-inverses 99.6%

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

   6. /-rgt-identity 99.6%

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

   7. +-commutative 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 3: 97.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2:\\ \;\;\;\;1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(1, 1 + x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.2) (+ 1.0 (/ 1.0 x)) (hypot 1.0 (+ 1.0 (* x 0.5)))))

C

double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = 1.0 + (1.0 / x);
	} else {
		tmp = hypot(1.0, (1.0 + (x * 0.5)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = 1.0 + (1.0 / x);
	} else {
		tmp = Math.hypot(1.0, (1.0 + (x * 0.5)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.2:
		tmp = 1.0 + (1.0 / x)
	else:
		tmp = math.hypot(1.0, (1.0 + (x * 0.5)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.2)
		tmp = Float64(1.0 + Float64(1.0 / x));
	else
		tmp = hypot(1.0, Float64(1.0 + Float64(x * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4.2)
		tmp = 1.0 + (1.0 / x);
	else
		tmp = hypot(1.0, (1.0 + (x * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.2], N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[Sqrt[1.0 ^ 2 + N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.20000000000000018

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. exp-lft-sqr 100.0%

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

      3. difference-of-sqr-1 100.0%

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

      4. associate-/l* 100.0%

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

      5. *-inverses 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative 100.0%

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

      2. flip-+ 100.0%

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

      3. metadata-eval 100.0%

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

      4. prod-exp 100.0%

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

      5. expm1-udef 100.0%

         \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{expm1}\left(x + x\right)}}{e^{x} - 1}} \]

      6. expm1-udef 100.0%

         \[\leadsto \sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\color{blue}{\mathsf{expm1}\left(x\right)}}} \]

      7. div-inv 100.0%

         \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(x + x\right) \cdot \frac{1}{\mathsf{expm1}\left(x\right)}}} \]

      8. div-inv 100.0%

         \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

      9. add-log-exp 100.0%

         \[\leadsto \color{blue}{\log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

      10. *-un-lft-identity 100.0%

          \[\leadsto \log \color{blue}{\left(1 \cdot e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

      11. log-prod 100.0%

          \[\leadsto \color{blue}{\log 1 + \log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

      12. metadata-eval 100.0%

          \[\leadsto \color{blue}{0} + \log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right) \]

      13. add-log-exp 100.0%

          \[\leadsto 0 + \color{blue}{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

      14. div-inv 100.0%

          \[\leadsto 0 + \sqrt{\color{blue}{\mathsf{expm1}\left(x + x\right) \cdot \frac{1}{\mathsf{expm1}\left(x\right)}}} \]

      15. div-inv 100.0%

          \[\leadsto 0 + \sqrt{\color{blue}{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

      16. expm1-udef 100.0%

          \[\leadsto 0 + \sqrt{\frac{\color{blue}{e^{x + x} - 1}}{\mathsf{expm1}\left(x\right)}} \]

      17. prod-exp 100.0%

          \[\leadsto 0 + \sqrt{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{\mathsf{expm1}\left(x\right)}} \]

      18. metadata-eval 100.0%

          \[\leadsto 0 + \sqrt{\frac{e^{x} \cdot e^{x} - \color{blue}{1 \cdot 1}}{\mathsf{expm1}\left(x\right)}} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \sqrt{1 + e^{x}}} \]
   6. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. rem-square-sqrt 100.0%

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

      3. hypot-1-def 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 5.2%

      \[\leadsto \mathsf{hypot}\left(1, \color{blue}{0.5 \cdot x + 1}\right) \]

   9. Taylor expanded in x around inf 1.1%

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

   10. Taylor expanded in x around 0 98.9%

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

   if -4.20000000000000018 < x

   1. Initial program 2.5%

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

      1. *-commutative 2.5%

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

      2. exp-lft-sqr 2.5%

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

      3. difference-of-sqr-1 2.8%

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

      4. associate-/l* 3.3%

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

      5. *-inverses 99.5%

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

      6. /-rgt-identity 99.5%

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

      7. +-commutative 99.5%

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

   3. Simplified 99.5%

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

      1. +-commutative 99.5%

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

      2. flip-+ 2.5%

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

      3. metadata-eval 2.5%

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

      4. prod-exp 2.5%

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

      5. expm1-udef 3.8%

         \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{expm1}\left(x + x\right)}}{e^{x} - 1}} \]

      6. expm1-udef 98.4%

         \[\leadsto \sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\color{blue}{\mathsf{expm1}\left(x\right)}}} \]

      7. div-inv 98.1%

         \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(x + x\right) \cdot \frac{1}{\mathsf{expm1}\left(x\right)}}} \]

      8. div-inv 98.4%

         \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

      9. add-log-exp 98.4%

         \[\leadsto \color{blue}{\log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

      10. *-un-lft-identity 98.4%

          \[\leadsto \log \color{blue}{\left(1 \cdot e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

      11. log-prod 98.4%

          \[\leadsto \color{blue}{\log 1 + \log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

      12. metadata-eval 98.4%

          \[\leadsto \color{blue}{0} + \log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right) \]

      13. add-log-exp 98.4%

          \[\leadsto 0 + \color{blue}{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

      14. div-inv 98.1%

          \[\leadsto 0 + \sqrt{\color{blue}{\mathsf{expm1}\left(x + x\right) \cdot \frac{1}{\mathsf{expm1}\left(x\right)}}} \]

      15. div-inv 98.4%

          \[\leadsto 0 + \sqrt{\color{blue}{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

      16. expm1-udef 5.1%

          \[\leadsto 0 + \sqrt{\frac{\color{blue}{e^{x + x} - 1}}{\mathsf{expm1}\left(x\right)}} \]

      17. prod-exp 5.1%

          \[\leadsto 0 + \sqrt{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{\mathsf{expm1}\left(x\right)}} \]

      18. metadata-eval 5.1%

          \[\leadsto 0 + \sqrt{\frac{e^{x} \cdot e^{x} - \color{blue}{1 \cdot 1}}{\mathsf{expm1}\left(x\right)}} \]

   5. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{0 + \sqrt{1 + e^{x}}} \]
   6. Step-by-step derivation

      1. +-lft-identity 99.5%

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

      2. rem-square-sqrt 99.5%

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

      3. hypot-1-def 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around 0 97.5%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2:\\ \;\;\;\;1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(1, 1 + x \cdot 0.5\right)\\ \end{array} \]

Alternative 4: 97.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8:\\ \;\;\;\;1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.8) (+ 1.0 (/ 1.0 x)) (sqrt (+ x 2.0))))

C

double code(double x) {
	double tmp;
	if (x <= -1.8) {
		tmp = 1.0 + (1.0 / x);
	} else {
		tmp = sqrt((x + 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.8d0)) then
        tmp = 1.0d0 + (1.0d0 / x)
    else
        tmp = sqrt((x + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.8) {
		tmp = 1.0 + (1.0 / x);
	} else {
		tmp = Math.sqrt((x + 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.8:
		tmp = 1.0 + (1.0 / x)
	else:
		tmp = math.sqrt((x + 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.8)
		tmp = Float64(1.0 + Float64(1.0 / x));
	else
		tmp = sqrt(Float64(x + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.8)
		tmp = 1.0 + (1.0 / x);
	else
		tmp = sqrt((x + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.8], N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(x + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.80000000000000004

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. exp-lft-sqr 100.0%

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

      3. difference-of-sqr-1 100.0%

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

      4. associate-/l* 100.0%

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

      5. *-inverses 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative 100.0%

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

      2. flip-+ 100.0%

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

      3. metadata-eval 100.0%

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

      4. prod-exp 100.0%

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

      5. expm1-udef 100.0%

         \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{expm1}\left(x + x\right)}}{e^{x} - 1}} \]

      6. expm1-udef 100.0%

         \[\leadsto \sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\color{blue}{\mathsf{expm1}\left(x\right)}}} \]

      7. div-inv 100.0%

         \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(x + x\right) \cdot \frac{1}{\mathsf{expm1}\left(x\right)}}} \]

      8. div-inv 100.0%

         \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

      9. add-log-exp 100.0%

         \[\leadsto \color{blue}{\log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

      10. *-un-lft-identity 100.0%

          \[\leadsto \log \color{blue}{\left(1 \cdot e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

      11. log-prod 100.0%

          \[\leadsto \color{blue}{\log 1 + \log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

      12. metadata-eval 100.0%

          \[\leadsto \color{blue}{0} + \log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right) \]

      13. add-log-exp 100.0%

          \[\leadsto 0 + \color{blue}{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

      14. div-inv 100.0%

          \[\leadsto 0 + \sqrt{\color{blue}{\mathsf{expm1}\left(x + x\right) \cdot \frac{1}{\mathsf{expm1}\left(x\right)}}} \]

      15. div-inv 100.0%

          \[\leadsto 0 + \sqrt{\color{blue}{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

      16. expm1-udef 100.0%

          \[\leadsto 0 + \sqrt{\frac{\color{blue}{e^{x + x} - 1}}{\mathsf{expm1}\left(x\right)}} \]

      17. prod-exp 100.0%

          \[\leadsto 0 + \sqrt{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{\mathsf{expm1}\left(x\right)}} \]

      18. metadata-eval 100.0%

          \[\leadsto 0 + \sqrt{\frac{e^{x} \cdot e^{x} - \color{blue}{1 \cdot 1}}{\mathsf{expm1}\left(x\right)}} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \sqrt{1 + e^{x}}} \]
   6. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. rem-square-sqrt 100.0%

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

      3. hypot-1-def 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 5.2%

      \[\leadsto \mathsf{hypot}\left(1, \color{blue}{0.5 \cdot x + 1}\right) \]

   9. Taylor expanded in x around inf 1.1%

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

   10. Taylor expanded in x around 0 98.9%

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

   if -1.80000000000000004 < x

   1. Initial program 2.5%

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

      1. *-commutative 2.5%

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

      2. exp-lft-sqr 2.5%

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

      3. difference-of-sqr-1 2.8%

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

      4. associate-/l* 3.3%

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

      5. *-inverses 99.5%

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

      6. /-rgt-identity 99.5%

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

      7. +-commutative 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 97.5%

      \[\leadsto \sqrt{\color{blue}{2 + x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8:\\ \;\;\;\;1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 2}\\ \end{array} \]

Alternative 5: 97.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x -5.0) (+ 1.0 (/ 1.0 x)) (sqrt 2.0)))

C

double code(double x) {
	double tmp;
	if (x <= -5.0) {
		tmp = 1.0 + (1.0 / x);
	} else {
		tmp = sqrt(2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5.0d0)) then
        tmp = 1.0d0 + (1.0d0 / x)
    else
        tmp = sqrt(2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -5.0) {
		tmp = 1.0 + (1.0 / x);
	} else {
		tmp = Math.sqrt(2.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5.0:
		tmp = 1.0 + (1.0 / x)
	else:
		tmp = math.sqrt(2.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5.0)
		tmp = Float64(1.0 + Float64(1.0 / x));
	else
		tmp = sqrt(2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5.0)
		tmp = 1.0 + (1.0 / x);
	else
		tmp = sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5.0], N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[Sqrt[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. exp-lft-sqr 100.0%

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

      3. difference-of-sqr-1 100.0%

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

      4. associate-/l* 100.0%

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

      5. *-inverses 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative 100.0%

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

      2. flip-+ 100.0%

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

      3. metadata-eval 100.0%

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

      4. prod-exp 100.0%

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

      5. expm1-udef 100.0%

         \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{expm1}\left(x + x\right)}}{e^{x} - 1}} \]

      6. expm1-udef 100.0%

         \[\leadsto \sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\color{blue}{\mathsf{expm1}\left(x\right)}}} \]

      7. div-inv 100.0%

         \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(x + x\right) \cdot \frac{1}{\mathsf{expm1}\left(x\right)}}} \]

      8. div-inv 100.0%

         \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

      9. add-log-exp 100.0%

         \[\leadsto \color{blue}{\log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

      10. *-un-lft-identity 100.0%

          \[\leadsto \log \color{blue}{\left(1 \cdot e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

      11. log-prod 100.0%

          \[\leadsto \color{blue}{\log 1 + \log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

      12. metadata-eval 100.0%

          \[\leadsto \color{blue}{0} + \log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right) \]

      13. add-log-exp 100.0%

          \[\leadsto 0 + \color{blue}{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

      14. div-inv 100.0%

          \[\leadsto 0 + \sqrt{\color{blue}{\mathsf{expm1}\left(x + x\right) \cdot \frac{1}{\mathsf{expm1}\left(x\right)}}} \]

      15. div-inv 100.0%

          \[\leadsto 0 + \sqrt{\color{blue}{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

      16. expm1-udef 100.0%

          \[\leadsto 0 + \sqrt{\frac{\color{blue}{e^{x + x} - 1}}{\mathsf{expm1}\left(x\right)}} \]

      17. prod-exp 100.0%

          \[\leadsto 0 + \sqrt{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{\mathsf{expm1}\left(x\right)}} \]

      18. metadata-eval 100.0%

          \[\leadsto 0 + \sqrt{\frac{e^{x} \cdot e^{x} - \color{blue}{1 \cdot 1}}{\mathsf{expm1}\left(x\right)}} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \sqrt{1 + e^{x}}} \]
   6. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. rem-square-sqrt 100.0%

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

      3. hypot-1-def 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 5.2%

      \[\leadsto \mathsf{hypot}\left(1, \color{blue}{0.5 \cdot x + 1}\right) \]

   9. Taylor expanded in x around inf 1.1%

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

   10. Taylor expanded in x around 0 98.9%

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

   if -5 < x

   1. Initial program 2.5%

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

      1. *-commutative 2.5%

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

      2. exp-lft-sqr 2.5%

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

      3. difference-of-sqr-1 2.8%

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

      4. associate-/l* 3.3%

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

      5. *-inverses 99.5%

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

      6. /-rgt-identity 99.5%

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

      7. +-commutative 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 97.0%

      \[\leadsto \color{blue}{\sqrt{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2}\\ \end{array} \]

Alternative 6: 46.0% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.4) (+ 1.0 (/ 1.0 x)) (+ 1.0 (* x 0.5))))

C

double code(double x) {
	double tmp;
	if (x <= -1.4) {
		tmp = 1.0 + (1.0 / x);
	} else {
		tmp = 1.0 + (x * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.4d0)) then
        tmp = 1.0d0 + (1.0d0 / x)
    else
        tmp = 1.0d0 + (x * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.4) {
		tmp = 1.0 + (1.0 / x);
	} else {
		tmp = 1.0 + (x * 0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.4:
		tmp = 1.0 + (1.0 / x)
	else:
		tmp = 1.0 + (x * 0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.4)
		tmp = Float64(1.0 + Float64(1.0 / x));
	else
		tmp = Float64(1.0 + Float64(x * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.4)
		tmp = 1.0 + (1.0 / x);
	else
		tmp = 1.0 + (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.4], N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. exp-lft-sqr 100.0%

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

      3. difference-of-sqr-1 100.0%

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

      4. associate-/l* 100.0%

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

      5. *-inverses 100.0%

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

      6. /-rgt-identity 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. +-commutative 100.0%

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

      2. flip-+ 100.0%

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

      3. metadata-eval 100.0%

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

      4. prod-exp 100.0%

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

      5. expm1-udef 100.0%

         \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{expm1}\left(x + x\right)}}{e^{x} - 1}} \]

      6. expm1-udef 100.0%

         \[\leadsto \sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\color{blue}{\mathsf{expm1}\left(x\right)}}} \]

      7. div-inv 100.0%

         \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(x + x\right) \cdot \frac{1}{\mathsf{expm1}\left(x\right)}}} \]

      8. div-inv 100.0%

         \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

      9. add-log-exp 100.0%

         \[\leadsto \color{blue}{\log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

      10. *-un-lft-identity 100.0%

          \[\leadsto \log \color{blue}{\left(1 \cdot e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

      11. log-prod 100.0%

          \[\leadsto \color{blue}{\log 1 + \log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

      12. metadata-eval 100.0%

          \[\leadsto \color{blue}{0} + \log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right) \]

      13. add-log-exp 100.0%

          \[\leadsto 0 + \color{blue}{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

      14. div-inv 100.0%

          \[\leadsto 0 + \sqrt{\color{blue}{\mathsf{expm1}\left(x + x\right) \cdot \frac{1}{\mathsf{expm1}\left(x\right)}}} \]

      15. div-inv 100.0%

          \[\leadsto 0 + \sqrt{\color{blue}{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

      16. expm1-udef 100.0%

          \[\leadsto 0 + \sqrt{\frac{\color{blue}{e^{x + x} - 1}}{\mathsf{expm1}\left(x\right)}} \]

      17. prod-exp 100.0%

          \[\leadsto 0 + \sqrt{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{\mathsf{expm1}\left(x\right)}} \]

      18. metadata-eval 100.0%

          \[\leadsto 0 + \sqrt{\frac{e^{x} \cdot e^{x} - \color{blue}{1 \cdot 1}}{\mathsf{expm1}\left(x\right)}} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \sqrt{1 + e^{x}}} \]
   6. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. rem-square-sqrt 100.0%

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

      3. hypot-1-def 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 5.2%

      \[\leadsto \mathsf{hypot}\left(1, \color{blue}{0.5 \cdot x + 1}\right) \]

   9. Taylor expanded in x around inf 1.1%

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

   10. Taylor expanded in x around 0 98.9%

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

   if -1.3999999999999999 < x

   1. Initial program 2.5%

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

      1. *-commutative 2.5%

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

      2. exp-lft-sqr 2.5%

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

      3. difference-of-sqr-1 2.8%

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

      4. associate-/l* 3.3%

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

      5. *-inverses 99.5%

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

      6. /-rgt-identity 99.5%

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

      7. +-commutative 99.5%

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

   3. Simplified 99.5%

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

      1. +-commutative 99.5%

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

      2. flip-+ 2.5%

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

      3. metadata-eval 2.5%

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

      4. prod-exp 2.5%

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

      5. expm1-udef 3.8%

         \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{expm1}\left(x + x\right)}}{e^{x} - 1}} \]

      6. expm1-udef 98.4%

         \[\leadsto \sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\color{blue}{\mathsf{expm1}\left(x\right)}}} \]

      7. div-inv 98.1%

         \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(x + x\right) \cdot \frac{1}{\mathsf{expm1}\left(x\right)}}} \]

      8. div-inv 98.4%

         \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

      9. add-log-exp 98.4%

         \[\leadsto \color{blue}{\log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

      10. *-un-lft-identity 98.4%

          \[\leadsto \log \color{blue}{\left(1 \cdot e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

      11. log-prod 98.4%

          \[\leadsto \color{blue}{\log 1 + \log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

      12. metadata-eval 98.4%

          \[\leadsto \color{blue}{0} + \log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right) \]

      13. add-log-exp 98.4%

          \[\leadsto 0 + \color{blue}{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

      14. div-inv 98.1%

          \[\leadsto 0 + \sqrt{\color{blue}{\mathsf{expm1}\left(x + x\right) \cdot \frac{1}{\mathsf{expm1}\left(x\right)}}} \]

      15. div-inv 98.4%

          \[\leadsto 0 + \sqrt{\color{blue}{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

      16. expm1-udef 5.1%

          \[\leadsto 0 + \sqrt{\frac{\color{blue}{e^{x + x} - 1}}{\mathsf{expm1}\left(x\right)}} \]

      17. prod-exp 5.1%

          \[\leadsto 0 + \sqrt{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{\mathsf{expm1}\left(x\right)}} \]

      18. metadata-eval 5.1%

          \[\leadsto 0 + \sqrt{\frac{e^{x} \cdot e^{x} - \color{blue}{1 \cdot 1}}{\mathsf{expm1}\left(x\right)}} \]

   5. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{0 + \sqrt{1 + e^{x}}} \]
   6. Step-by-step derivation

      1. +-lft-identity 99.5%

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

      2. rem-square-sqrt 99.5%

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

      3. hypot-1-def 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around 0 97.5%

      \[\leadsto \mathsf{hypot}\left(1, \color{blue}{0.5 \cdot x + 1}\right) \]

   9. Taylor expanded in x around inf 20.5%

      \[\leadsto \color{blue}{1 + 0.5 \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot 0.5\\ \end{array} \]

Alternative 7: 14.1% accurate, 61.8× speedup

Math

\[\begin{array}{l} \\ 1 + x \cdot 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 1.0 (* x 0.5)))

C

double code(double x) {
	return 1.0 + (x * 0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + (x * 0.5d0)
end function

Java

public static double code(double x) {
	return 1.0 + (x * 0.5);
}

Python

def code(x):
	return 1.0 + (x * 0.5)

Julia

function code(x)
	return Float64(1.0 + Float64(x * 0.5))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 + (x * 0.5);
end

Wolfram

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

Derivation

1. Initial program 30.7%

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

   1. *-commutative 30.7%

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

   2. exp-lft-sqr 30.7%

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

   3. difference-of-sqr-1 30.9%

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

   4. associate-/l* 31.2%

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

   5. *-inverses 99.6%

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

   6. /-rgt-identity 99.6%

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

   7. +-commutative 99.6%

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

3. Simplified 99.6%

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

   1. +-commutative 99.6%

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

   2. flip-+ 30.7%

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

   3. metadata-eval 30.7%

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

   4. prod-exp 30.7%

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

   5. expm1-udef 31.6%

      \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{expm1}\left(x + x\right)}}{e^{x} - 1}} \]

   6. expm1-udef 98.8%

      \[\leadsto \sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\color{blue}{\mathsf{expm1}\left(x\right)}}} \]

   7. div-inv 98.7%

      \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(x + x\right) \cdot \frac{1}{\mathsf{expm1}\left(x\right)}}} \]

   8. div-inv 98.8%

      \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

   9. add-log-exp 98.8%

      \[\leadsto \color{blue}{\log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

   10. *-un-lft-identity 98.8%

       \[\leadsto \log \color{blue}{\left(1 \cdot e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

   11. log-prod 98.8%

       \[\leadsto \color{blue}{\log 1 + \log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

   12. metadata-eval 98.8%

       \[\leadsto \color{blue}{0} + \log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right) \]

   13. add-log-exp 98.8%

       \[\leadsto 0 + \color{blue}{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

   14. div-inv 98.7%

       \[\leadsto 0 + \sqrt{\color{blue}{\mathsf{expm1}\left(x + x\right) \cdot \frac{1}{\mathsf{expm1}\left(x\right)}}} \]

   15. div-inv 98.8%

       \[\leadsto 0 + \sqrt{\color{blue}{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

   16. expm1-udef 32.6%

       \[\leadsto 0 + \sqrt{\frac{\color{blue}{e^{x + x} - 1}}{\mathsf{expm1}\left(x\right)}} \]

   17. prod-exp 32.6%

       \[\leadsto 0 + \sqrt{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{\mathsf{expm1}\left(x\right)}} \]

   18. metadata-eval 32.6%

       \[\leadsto 0 + \sqrt{\frac{e^{x} \cdot e^{x} - \color{blue}{1 \cdot 1}}{\mathsf{expm1}\left(x\right)}} \]

5. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{0 + \sqrt{1 + e^{x}}} \]
6. Step-by-step derivation

   1. +-lft-identity 99.6%

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

   2. rem-square-sqrt 99.6%

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

   3. hypot-1-def 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in x around 0 70.8%

   \[\leadsto \mathsf{hypot}\left(1, \color{blue}{0.5 \cdot x + 1}\right) \]

9. Taylor expanded in x around inf 14.9%

   \[\leadsto \color{blue}{1 + 0.5 \cdot x} \]

10. Final simplification 14.9%

    \[\leadsto 1 + x \cdot 0.5 \]

Alternative 8: 2.9% accurate, 103.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 0.5))

C

double code(double x) {
	return x * 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * 0.5d0
end function

Java

public static double code(double x) {
	return x * 0.5;
}

Python

def code(x):
	return x * 0.5

Julia

function code(x)
	return Float64(x * 0.5)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.5;
end

Wolfram

code[x_] := N[(x * 0.5), $MachinePrecision]

Derivation

1. Initial program 30.7%

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

   1. *-commutative 30.7%

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

   2. exp-lft-sqr 30.7%

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

   3. difference-of-sqr-1 30.9%

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

   4. associate-/l* 31.2%

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

   5. *-inverses 99.6%

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

   6. /-rgt-identity 99.6%

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

   7. +-commutative 99.6%

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

3. Simplified 99.6%

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

   1. +-commutative 99.6%

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

   2. flip-+ 30.7%

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

   3. metadata-eval 30.7%

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

   4. prod-exp 30.7%

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

   5. expm1-udef 31.6%

      \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{expm1}\left(x + x\right)}}{e^{x} - 1}} \]

   6. expm1-udef 98.8%

      \[\leadsto \sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\color{blue}{\mathsf{expm1}\left(x\right)}}} \]

   7. div-inv 98.7%

      \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(x + x\right) \cdot \frac{1}{\mathsf{expm1}\left(x\right)}}} \]

   8. div-inv 98.8%

      \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

   9. add-log-exp 98.8%

      \[\leadsto \color{blue}{\log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

   10. *-un-lft-identity 98.8%

       \[\leadsto \log \color{blue}{\left(1 \cdot e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

   11. log-prod 98.8%

       \[\leadsto \color{blue}{\log 1 + \log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

   12. metadata-eval 98.8%

       \[\leadsto \color{blue}{0} + \log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right) \]

   13. add-log-exp 98.8%

       \[\leadsto 0 + \color{blue}{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

   14. div-inv 98.7%

       \[\leadsto 0 + \sqrt{\color{blue}{\mathsf{expm1}\left(x + x\right) \cdot \frac{1}{\mathsf{expm1}\left(x\right)}}} \]

   15. div-inv 98.8%

       \[\leadsto 0 + \sqrt{\color{blue}{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

   16. expm1-udef 32.6%

       \[\leadsto 0 + \sqrt{\frac{\color{blue}{e^{x + x} - 1}}{\mathsf{expm1}\left(x\right)}} \]

   17. prod-exp 32.6%

       \[\leadsto 0 + \sqrt{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{\mathsf{expm1}\left(x\right)}} \]

   18. metadata-eval 32.6%

       \[\leadsto 0 + \sqrt{\frac{e^{x} \cdot e^{x} - \color{blue}{1 \cdot 1}}{\mathsf{expm1}\left(x\right)}} \]

5. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{0 + \sqrt{1 + e^{x}}} \]
6. Step-by-step derivation

   1. +-lft-identity 99.6%

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

   2. rem-square-sqrt 99.6%

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

   3. hypot-1-def 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in x around 0 70.8%

   \[\leadsto \mathsf{hypot}\left(1, \color{blue}{0.5 \cdot x + 1}\right) \]

9. Taylor expanded in x around inf 3.3%

   \[\leadsto \color{blue}{0.5 \cdot x} \]

10. Final simplification 3.3%

    \[\leadsto x \cdot 0.5 \]

Alternative 9: 4.3% accurate, 103.0× speedup

Math

\[\begin{array}{l} \\ x \cdot -0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x -0.5))

C

double code(double x) {
	return x * -0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (-0.5d0)
end function

Java

public static double code(double x) {
	return x * -0.5;
}

Python

def code(x):
	return x * -0.5

Julia

function code(x)
	return Float64(x * -0.5)
end

MATLAB

function tmp = code(x)
	tmp = x * -0.5;
end

Wolfram

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

Derivation

1. Initial program 30.7%

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

   1. *-commutative 30.7%

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

   2. exp-lft-sqr 30.7%

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

   3. difference-of-sqr-1 30.9%

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

   4. associate-/l* 31.2%

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

   5. *-inverses 99.6%

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

   6. /-rgt-identity 99.6%

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

   7. +-commutative 99.6%

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

3. Simplified 99.6%

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

   1. +-commutative 99.6%

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

   2. flip-+ 30.7%

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

   3. metadata-eval 30.7%

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

   4. prod-exp 30.7%

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

   5. expm1-udef 31.6%

      \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{expm1}\left(x + x\right)}}{e^{x} - 1}} \]

   6. expm1-udef 98.8%

      \[\leadsto \sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\color{blue}{\mathsf{expm1}\left(x\right)}}} \]

   7. div-inv 98.7%

      \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(x + x\right) \cdot \frac{1}{\mathsf{expm1}\left(x\right)}}} \]

   8. div-inv 98.8%

      \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

   9. add-log-exp 98.8%

      \[\leadsto \color{blue}{\log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

   10. *-un-lft-identity 98.8%

       \[\leadsto \log \color{blue}{\left(1 \cdot e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

   11. log-prod 98.8%

       \[\leadsto \color{blue}{\log 1 + \log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right)} \]

   12. metadata-eval 98.8%

       \[\leadsto \color{blue}{0} + \log \left(e^{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}}\right) \]

   13. add-log-exp 98.8%

       \[\leadsto 0 + \color{blue}{\sqrt{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

   14. div-inv 98.7%

       \[\leadsto 0 + \sqrt{\color{blue}{\mathsf{expm1}\left(x + x\right) \cdot \frac{1}{\mathsf{expm1}\left(x\right)}}} \]

   15. div-inv 98.8%

       \[\leadsto 0 + \sqrt{\color{blue}{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}} \]

   16. expm1-udef 32.6%

       \[\leadsto 0 + \sqrt{\frac{\color{blue}{e^{x + x} - 1}}{\mathsf{expm1}\left(x\right)}} \]

   17. prod-exp 32.6%

       \[\leadsto 0 + \sqrt{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{\mathsf{expm1}\left(x\right)}} \]

   18. metadata-eval 32.6%

       \[\leadsto 0 + \sqrt{\frac{e^{x} \cdot e^{x} - \color{blue}{1 \cdot 1}}{\mathsf{expm1}\left(x\right)}} \]

5. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{0 + \sqrt{1 + e^{x}}} \]
6. Step-by-step derivation

   1. +-lft-identity 99.6%

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

   2. rem-square-sqrt 99.6%

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

   3. hypot-1-def 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in x around 0 70.8%

   \[\leadsto \mathsf{hypot}\left(1, \color{blue}{0.5 \cdot x + 1}\right) \]

9. Taylor expanded in x around -inf 3.9%

   \[\leadsto \color{blue}{-0.5 \cdot x} \]
10. Step-by-step derivation

    1. *-commutative 3.9%

       \[\leadsto \color{blue}{x \cdot -0.5} \]

11. Simplified 3.9%

    \[\leadsto \color{blue}{x \cdot -0.5} \]

12. Final simplification 3.9%

    \[\leadsto x \cdot -0.5 \]

Reproduce

herbie shell --seed 2023187 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))