exp2 (problem 3.3.7)

Percentage Accurate: 76.6% → 99.8%

Time: 5.1s

Alternatives: 7

Speedup: 2.0×

• Unsound rule application detected in e-graph. Results may not be sound.

• 49.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(e^{x} - 2\right) + e^{-x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (- (exp x) 2.0) (exp (- x))))

C

double code(double x) {
	return (exp(x) - 2.0) + exp(-x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (exp(x) - 2.0d0) + exp(-x)
end function

Java

public static double code(double x) {
	return (Math.exp(x) - 2.0) + Math.exp(-x);
}

Python

def code(x):
	return (math.exp(x) - 2.0) + math.exp(-x)

Julia

function code(x)
	return Float64(Float64(exp(x) - 2.0) + exp(Float64(-x)))
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - 2.0) + exp(-x);
end

Wolfram

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]

Initial Program: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(e^{x} - 2\right) + e^{-x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (- (exp x) 2.0) (exp (- x))))

C

double code(double x) {
	return (exp(x) - 2.0) + exp(-x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (exp(x) - 2.0d0) + exp(-x)
end function

Java

public static double code(double x) {
	return (Math.exp(x) - 2.0) + Math.exp(-x);
}

Python

def code(x):
	return (math.exp(x) - 2.0) + math.exp(-x)

Julia

function code(x)
	return Float64(Float64(exp(x) - 2.0) + exp(Float64(-x)))
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - 2.0) + exp(-x);
end

Wolfram

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(e^{x} - 2\right) + t_0 \leq 2 \cdot 10^{-11}:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(t_0 + -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (+ (- (exp x) 2.0) t_0) 2e-11)
     (+ (* x x) (* 0.08333333333333333 (pow x 4.0)))
     (+ (exp x) (+ t_0 -2.0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (((exp(x) - 2.0) + t_0) <= 2e-11) {
		tmp = (x * x) + (0.08333333333333333 * pow(x, 4.0));
	} else {
		tmp = exp(x) + (t_0 + -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (((exp(x) - 2.0d0) + t_0) <= 2d-11) then
        tmp = (x * x) + (0.08333333333333333d0 * (x ** 4.0d0))
    else
        tmp = exp(x) + (t_0 + (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (((Math.exp(x) - 2.0) + t_0) <= 2e-11) {
		tmp = (x * x) + (0.08333333333333333 * Math.pow(x, 4.0));
	} else {
		tmp = Math.exp(x) + (t_0 + -2.0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.exp(-x)
	tmp = 0
	if ((math.exp(x) - 2.0) + t_0) <= 2e-11:
		tmp = (x * x) + (0.08333333333333333 * math.pow(x, 4.0))
	else:
		tmp = math.exp(x) + (t_0 + -2.0)
	return tmp

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(Float64(exp(x) - 2.0) + t_0) <= 2e-11)
		tmp = Float64(Float64(x * x) + Float64(0.08333333333333333 * (x ^ 4.0)));
	else
		tmp = Float64(exp(x) + Float64(t_0 + -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = exp(-x);
	tmp = 0.0;
	if (((exp(x) - 2.0) + t_0) <= 2e-11)
		tmp = (x * x) + (0.08333333333333333 * (x ^ 4.0));
	else
		tmp = exp(x) + (t_0 + -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + t$95$0), $MachinePrecision], 2e-11], N[(N[(x * x), $MachinePrecision] + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[x], $MachinePrecision] + N[(t$95$0 + -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 1.99999999999999988e-11

   1. Initial program 52.8%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Step-by-step derivation

      1. associate-+l- 53.0%

         \[\leadsto \color{blue}{e^{x} - \left(2 - e^{-x}\right)} \]

      2. sub-neg 53.0%

         \[\leadsto \color{blue}{e^{x} + \left(-\left(2 - e^{-x}\right)\right)} \]

      3. sub-neg 53.0%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(2 + \left(-e^{-x}\right)\right)}\right) \]

      4. +-commutative 53.0%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(\left(-e^{-x}\right) + 2\right)}\right) \]

      5. distribute-neg-in 53.0%

         \[\leadsto e^{x} + \color{blue}{\left(\left(-\left(-e^{-x}\right)\right) + \left(-2\right)\right)} \]

      6. remove-double-neg 53.0%

         \[\leadsto e^{x} + \left(\color{blue}{e^{-x}} + \left(-2\right)\right) \]

      7. metadata-eval 53.0%

         \[\leadsto e^{x} + \left(e^{-x} + \color{blue}{-2}\right) \]

   3. Simplified 53.0%

      \[\leadsto \color{blue}{e^{x} + \left(e^{-x} + -2\right)} \]

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{x}^{2} + 0.08333333333333333 \cdot {x}^{4}} \]
   5. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot x + 0.08333333333333333 \cdot {x}^{4}} \]

   if 1.99999999999999988e-11 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 99.9%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Step-by-step derivation

      1. associate-+l- 99.9%

         \[\leadsto \color{blue}{e^{x} - \left(2 - e^{-x}\right)} \]

      2. sub-neg 99.9%

         \[\leadsto \color{blue}{e^{x} + \left(-\left(2 - e^{-x}\right)\right)} \]

      3. sub-neg 99.9%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(2 + \left(-e^{-x}\right)\right)}\right) \]

      4. +-commutative 99.9%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(\left(-e^{-x}\right) + 2\right)}\right) \]

      5. distribute-neg-in 99.9%

         \[\leadsto e^{x} + \color{blue}{\left(\left(-\left(-e^{-x}\right)\right) + \left(-2\right)\right)} \]

      6. remove-double-neg 99.9%

         \[\leadsto e^{x} + \left(\color{blue}{e^{-x}} + \left(-2\right)\right) \]

      7. metadata-eval 99.9%

         \[\leadsto e^{x} + \left(e^{-x} + \color{blue}{-2}\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{e^{x} + \left(e^{-x} + -2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(e^{-x} + -2\right)\\ \end{array} \]

Alternative 2: 96.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{0.006944444444444444 \cdot {x}^{8}}\\ \mathbf{elif}\;x \leq 2.6:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e+39)
   (sqrt (* 0.006944444444444444 (pow x 8.0)))
   (if (<= x 2.6) (+ (* x x) (* 0.08333333333333333 (pow x 4.0))) (expm1 x))))

C

double code(double x) {
	double tmp;
	if (x <= -5e+39) {
		tmp = sqrt((0.006944444444444444 * pow(x, 8.0)));
	} else if (x <= 2.6) {
		tmp = (x * x) + (0.08333333333333333 * pow(x, 4.0));
	} else {
		tmp = expm1(x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -5e+39) {
		tmp = Math.sqrt((0.006944444444444444 * Math.pow(x, 8.0)));
	} else if (x <= 2.6) {
		tmp = (x * x) + (0.08333333333333333 * Math.pow(x, 4.0));
	} else {
		tmp = Math.expm1(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5e+39:
		tmp = math.sqrt((0.006944444444444444 * math.pow(x, 8.0)))
	elif x <= 2.6:
		tmp = (x * x) + (0.08333333333333333 * math.pow(x, 4.0))
	else:
		tmp = math.expm1(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e+39)
		tmp = sqrt(Float64(0.006944444444444444 * (x ^ 8.0)));
	elseif (x <= 2.6)
		tmp = Float64(Float64(x * x) + Float64(0.08333333333333333 * (x ^ 4.0)));
	else
		tmp = expm1(x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e+39], N[Sqrt[N[(0.006944444444444444 * N[Power[x, 8.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2.6], N[(N[(x * x), $MachinePrecision] + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Exp[x] - 1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.00000000000000015e39

   1. Initial program 100.0%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{e^{x} - \left(2 - e^{-x}\right)} \]

      2. sub-neg 100.0%

         \[\leadsto \color{blue}{e^{x} + \left(-\left(2 - e^{-x}\right)\right)} \]

      3. sub-neg 100.0%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(2 + \left(-e^{-x}\right)\right)}\right) \]

      4. +-commutative 100.0%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(\left(-e^{-x}\right) + 2\right)}\right) \]

      5. distribute-neg-in 100.0%

         \[\leadsto e^{x} + \color{blue}{\left(\left(-\left(-e^{-x}\right)\right) + \left(-2\right)\right)} \]

      6. remove-double-neg 100.0%

         \[\leadsto e^{x} + \left(\color{blue}{e^{-x}} + \left(-2\right)\right) \]

      7. metadata-eval 100.0%

         \[\leadsto e^{x} + \left(e^{-x} + \color{blue}{-2}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x} + \left(e^{-x} + -2\right)} \]

   4. Taylor expanded in x around 0 88.1%

      \[\leadsto \color{blue}{{x}^{2} + 0.08333333333333333 \cdot {x}^{4}} \]
   5. Step-by-step derivation

      1. unpow2 88.1%

         \[\leadsto \color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4} \]

   6. Simplified 88.1%

      \[\leadsto \color{blue}{x \cdot x + 0.08333333333333333 \cdot {x}^{4}} \]

   7. Taylor expanded in x around inf 88.1%

      \[\leadsto \color{blue}{0.08333333333333333 \cdot {x}^{4}} \]
   8. Step-by-step derivation

      1. *-commutative 88.1%

         \[\leadsto \color{blue}{{x}^{4} \cdot 0.08333333333333333} \]

   9. Simplified 88.1%

      \[\leadsto \color{blue}{{x}^{4} \cdot 0.08333333333333333} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 88.1%

          \[\leadsto \color{blue}{\sqrt{{x}^{4} \cdot 0.08333333333333333} \cdot \sqrt{{x}^{4} \cdot 0.08333333333333333}} \]

       2. sqrt-unprod 100.0%

          \[\leadsto \color{blue}{\sqrt{\left({x}^{4} \cdot 0.08333333333333333\right) \cdot \left({x}^{4} \cdot 0.08333333333333333\right)}} \]

       3. pow1/2 100.0%

          \[\leadsto \color{blue}{{\left(\left({x}^{4} \cdot 0.08333333333333333\right) \cdot \left({x}^{4} \cdot 0.08333333333333333\right)\right)}^{0.5}} \]

       4. *-commutative 100.0%

          \[\leadsto {\left(\color{blue}{\left(0.08333333333333333 \cdot {x}^{4}\right)} \cdot \left({x}^{4} \cdot 0.08333333333333333\right)\right)}^{0.5} \]

       5. *-commutative 100.0%

          \[\leadsto {\left(\left(0.08333333333333333 \cdot {x}^{4}\right) \cdot \color{blue}{\left(0.08333333333333333 \cdot {x}^{4}\right)}\right)}^{0.5} \]

       6. swap-sqr 100.0%

          \[\leadsto {\color{blue}{\left(\left(0.08333333333333333 \cdot 0.08333333333333333\right) \cdot \left({x}^{4} \cdot {x}^{4}\right)\right)}}^{0.5} \]

       7. metadata-eval 100.0%

          \[\leadsto {\left(\color{blue}{0.006944444444444444} \cdot \left({x}^{4} \cdot {x}^{4}\right)\right)}^{0.5} \]

       8. pow-prod-up 100.0%

          \[\leadsto {\left(0.006944444444444444 \cdot \color{blue}{{x}^{\left(4 + 4\right)}}\right)}^{0.5} \]

       9. metadata-eval 100.0%

          \[\leadsto {\left(0.006944444444444444 \cdot {x}^{\color{blue}{8}}\right)}^{0.5} \]

   11. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{{\left(0.006944444444444444 \cdot {x}^{8}\right)}^{0.5}} \]
   12. Step-by-step derivation

       1. unpow1/2 100.0%

          \[\leadsto \color{blue}{\sqrt{0.006944444444444444 \cdot {x}^{8}}} \]

   13. Simplified 100.0%

       \[\leadsto \color{blue}{\sqrt{0.006944444444444444 \cdot {x}^{8}}} \]

   if -5.00000000000000015e39 < x < 2.60000000000000009

   1. Initial program 55.9%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Step-by-step derivation

      1. associate-+l- 56.1%

         \[\leadsto \color{blue}{e^{x} - \left(2 - e^{-x}\right)} \]

      2. sub-neg 56.1%

         \[\leadsto \color{blue}{e^{x} + \left(-\left(2 - e^{-x}\right)\right)} \]

      3. sub-neg 56.1%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(2 + \left(-e^{-x}\right)\right)}\right) \]

      4. +-commutative 56.1%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(\left(-e^{-x}\right) + 2\right)}\right) \]

      5. distribute-neg-in 56.1%

         \[\leadsto e^{x} + \color{blue}{\left(\left(-\left(-e^{-x}\right)\right) + \left(-2\right)\right)} \]

      6. remove-double-neg 56.1%

         \[\leadsto e^{x} + \left(\color{blue}{e^{-x}} + \left(-2\right)\right) \]

      7. metadata-eval 56.1%

         \[\leadsto e^{x} + \left(e^{-x} + \color{blue}{-2}\right) \]

   3. Simplified 56.1%

      \[\leadsto \color{blue}{e^{x} + \left(e^{-x} + -2\right)} \]

   4. Taylor expanded in x around 0 94.0%

      \[\leadsto \color{blue}{{x}^{2} + 0.08333333333333333 \cdot {x}^{4}} \]
   5. Step-by-step derivation

      1. unpow2 94.0%

         \[\leadsto \color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4} \]

   6. Simplified 94.0%

      \[\leadsto \color{blue}{x \cdot x + 0.08333333333333333 \cdot {x}^{4}} \]

   if 2.60000000000000009 < x

   1. Initial program 100.0%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{e^{x} - \left(2 - e^{-x}\right)} \]

      2. sub-neg 100.0%

         \[\leadsto \color{blue}{e^{x} + \left(-\left(2 - e^{-x}\right)\right)} \]

      3. sub-neg 100.0%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(2 + \left(-e^{-x}\right)\right)}\right) \]

      4. +-commutative 100.0%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(\left(-e^{-x}\right) + 2\right)}\right) \]

      5. distribute-neg-in 100.0%

         \[\leadsto e^{x} + \color{blue}{\left(\left(-\left(-e^{-x}\right)\right) + \left(-2\right)\right)} \]

      6. remove-double-neg 100.0%

         \[\leadsto e^{x} + \left(\color{blue}{e^{-x}} + \left(-2\right)\right) \]

      7. metadata-eval 100.0%

         \[\leadsto e^{x} + \left(e^{-x} + \color{blue}{-2}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x} + \left(e^{-x} + -2\right)} \]

   4. Taylor expanded in x around 0 98.8%

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

   5. Taylor expanded in x around inf 98.8%

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

      1. expm1-def 98.8%

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

   7. Simplified 98.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{0.006944444444444444 \cdot {x}^{8}}\\ \mathbf{elif}\;x \leq 2.6:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 3: 93.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.6) (+ (* x x) (* 0.08333333333333333 (pow x 4.0))) (expm1 x)))

C

double code(double x) {
	double tmp;
	if (x <= 2.6) {
		tmp = (x * x) + (0.08333333333333333 * pow(x, 4.0));
	} else {
		tmp = expm1(x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.6) {
		tmp = (x * x) + (0.08333333333333333 * Math.pow(x, 4.0));
	} else {
		tmp = Math.expm1(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.6:
		tmp = (x * x) + (0.08333333333333333 * math.pow(x, 4.0))
	else:
		tmp = math.expm1(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.6)
		tmp = Float64(Float64(x * x) + Float64(0.08333333333333333 * (x ^ 4.0)));
	else
		tmp = expm1(x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.6], N[(N[(x * x), $MachinePrecision] + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Exp[x] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.60000000000000009

   1. Initial program 68.5%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Step-by-step derivation

      1. associate-+l- 68.7%

         \[\leadsto \color{blue}{e^{x} - \left(2 - e^{-x}\right)} \]

      2. sub-neg 68.7%

         \[\leadsto \color{blue}{e^{x} + \left(-\left(2 - e^{-x}\right)\right)} \]

      3. sub-neg 68.7%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(2 + \left(-e^{-x}\right)\right)}\right) \]

      4. +-commutative 68.7%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(\left(-e^{-x}\right) + 2\right)}\right) \]

      5. distribute-neg-in 68.7%

         \[\leadsto e^{x} + \color{blue}{\left(\left(-\left(-e^{-x}\right)\right) + \left(-2\right)\right)} \]

      6. remove-double-neg 68.7%

         \[\leadsto e^{x} + \left(\color{blue}{e^{-x}} + \left(-2\right)\right) \]

      7. metadata-eval 68.7%

         \[\leadsto e^{x} + \left(e^{-x} + \color{blue}{-2}\right) \]

   3. Simplified 68.7%

      \[\leadsto \color{blue}{e^{x} + \left(e^{-x} + -2\right)} \]

   4. Taylor expanded in x around 0 92.3%

      \[\leadsto \color{blue}{{x}^{2} + 0.08333333333333333 \cdot {x}^{4}} \]
   5. Step-by-step derivation

      1. unpow2 92.3%

         \[\leadsto \color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4} \]

   6. Simplified 92.3%

      \[\leadsto \color{blue}{x \cdot x + 0.08333333333333333 \cdot {x}^{4}} \]

   if 2.60000000000000009 < x

   1. Initial program 100.0%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{e^{x} - \left(2 - e^{-x}\right)} \]

      2. sub-neg 100.0%

         \[\leadsto \color{blue}{e^{x} + \left(-\left(2 - e^{-x}\right)\right)} \]

      3. sub-neg 100.0%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(2 + \left(-e^{-x}\right)\right)}\right) \]

      4. +-commutative 100.0%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(\left(-e^{-x}\right) + 2\right)}\right) \]

      5. distribute-neg-in 100.0%

         \[\leadsto e^{x} + \color{blue}{\left(\left(-\left(-e^{-x}\right)\right) + \left(-2\right)\right)} \]

      6. remove-double-neg 100.0%

         \[\leadsto e^{x} + \left(\color{blue}{e^{-x}} + \left(-2\right)\right) \]

      7. metadata-eval 100.0%

         \[\leadsto e^{x} + \left(e^{-x} + \color{blue}{-2}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x} + \left(e^{-x} + -2\right)} \]

   4. Taylor expanded in x around 0 98.8%

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

   5. Taylor expanded in x around inf 98.8%

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

      1. expm1-def 98.8%

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

   7. Simplified 98.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 4: 93.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.45:\\ \;\;\;\;0.08333333333333333 \cdot {x}^{4}\\ \mathbf{elif}\;x \leq 1.65:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.45)
   (* 0.08333333333333333 (pow x 4.0))
   (if (<= x 1.65) (* x x) (expm1 x))))

C

double code(double x) {
	double tmp;
	if (x <= -3.45) {
		tmp = 0.08333333333333333 * pow(x, 4.0);
	} else if (x <= 1.65) {
		tmp = x * x;
	} else {
		tmp = expm1(x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.45) {
		tmp = 0.08333333333333333 * Math.pow(x, 4.0);
	} else if (x <= 1.65) {
		tmp = x * x;
	} else {
		tmp = Math.expm1(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.45:
		tmp = 0.08333333333333333 * math.pow(x, 4.0)
	elif x <= 1.65:
		tmp = x * x
	else:
		tmp = math.expm1(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.45)
		tmp = Float64(0.08333333333333333 * (x ^ 4.0));
	elseif (x <= 1.65)
		tmp = Float64(x * x);
	else
		tmp = expm1(x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -3.45], N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.65], N[(x * x), $MachinePrecision], N[(Exp[x] - 1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4500000000000002

   1. Initial program 100.0%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{e^{x} - \left(2 - e^{-x}\right)} \]

      2. sub-neg 100.0%

         \[\leadsto \color{blue}{e^{x} + \left(-\left(2 - e^{-x}\right)\right)} \]

      3. sub-neg 100.0%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(2 + \left(-e^{-x}\right)\right)}\right) \]

      4. +-commutative 100.0%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(\left(-e^{-x}\right) + 2\right)}\right) \]

      5. distribute-neg-in 100.0%

         \[\leadsto e^{x} + \color{blue}{\left(\left(-\left(-e^{-x}\right)\right) + \left(-2\right)\right)} \]

      6. remove-double-neg 100.0%

         \[\leadsto e^{x} + \left(\color{blue}{e^{-x}} + \left(-2\right)\right) \]

      7. metadata-eval 100.0%

         \[\leadsto e^{x} + \left(e^{-x} + \color{blue}{-2}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x} + \left(e^{-x} + -2\right)} \]

   4. Taylor expanded in x around 0 77.4%

      \[\leadsto \color{blue}{{x}^{2} + 0.08333333333333333 \cdot {x}^{4}} \]
   5. Step-by-step derivation

      1. unpow2 77.4%

         \[\leadsto \color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4} \]

   6. Simplified 77.4%

      \[\leadsto \color{blue}{x \cdot x + 0.08333333333333333 \cdot {x}^{4}} \]

   7. Taylor expanded in x around inf 77.4%

      \[\leadsto \color{blue}{0.08333333333333333 \cdot {x}^{4}} \]
   8. Step-by-step derivation

      1. *-commutative 77.4%

         \[\leadsto \color{blue}{{x}^{4} \cdot 0.08333333333333333} \]

   9. Simplified 77.4%

      \[\leadsto \color{blue}{{x}^{4} \cdot 0.08333333333333333} \]

   if -3.4500000000000002 < x < 1.6499999999999999

   1. Initial program 53.2%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Step-by-step derivation

      1. associate-+l- 53.3%

         \[\leadsto \color{blue}{e^{x} - \left(2 - e^{-x}\right)} \]

      2. sub-neg 53.3%

         \[\leadsto \color{blue}{e^{x} + \left(-\left(2 - e^{-x}\right)\right)} \]

      3. sub-neg 53.3%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(2 + \left(-e^{-x}\right)\right)}\right) \]

      4. +-commutative 53.3%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(\left(-e^{-x}\right) + 2\right)}\right) \]

      5. distribute-neg-in 53.3%

         \[\leadsto e^{x} + \color{blue}{\left(\left(-\left(-e^{-x}\right)\right) + \left(-2\right)\right)} \]

      6. remove-double-neg 53.3%

         \[\leadsto e^{x} + \left(\color{blue}{e^{-x}} + \left(-2\right)\right) \]

      7. metadata-eval 53.3%

         \[\leadsto e^{x} + \left(e^{-x} + \color{blue}{-2}\right) \]

   3. Simplified 53.3%

      \[\leadsto \color{blue}{e^{x} + \left(e^{-x} + -2\right)} \]

   4. Taylor expanded in x around 0 99.2%

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

      1. unpow2 99.2%

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

   6. Simplified 99.2%

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

   if 1.6499999999999999 < x

   1. Initial program 100.0%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{e^{x} - \left(2 - e^{-x}\right)} \]

      2. sub-neg 100.0%

         \[\leadsto \color{blue}{e^{x} + \left(-\left(2 - e^{-x}\right)\right)} \]

      3. sub-neg 100.0%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(2 + \left(-e^{-x}\right)\right)}\right) \]

      4. +-commutative 100.0%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(\left(-e^{-x}\right) + 2\right)}\right) \]

      5. distribute-neg-in 100.0%

         \[\leadsto e^{x} + \color{blue}{\left(\left(-\left(-e^{-x}\right)\right) + \left(-2\right)\right)} \]

      6. remove-double-neg 100.0%

         \[\leadsto e^{x} + \left(\color{blue}{e^{-x}} + \left(-2\right)\right) \]

      7. metadata-eval 100.0%

         \[\leadsto e^{x} + \left(e^{-x} + \color{blue}{-2}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x} + \left(e^{-x} + -2\right)} \]

   4. Taylor expanded in x around 0 98.8%

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

   5. Taylor expanded in x around inf 98.8%

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

      1. expm1-def 98.8%

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

   7. Simplified 98.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.45:\\ \;\;\;\;0.08333333333333333 \cdot {x}^{4}\\ \mathbf{elif}\;x \leq 1.65:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 5: 87.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.65:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.65) (* x x) (expm1 x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.65) {
		tmp = x * x;
	} else {
		tmp = expm1(x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.65) {
		tmp = x * x;
	} else {
		tmp = Math.expm1(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.65:
		tmp = x * x
	else:
		tmp = math.expm1(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.65)
		tmp = Float64(x * x);
	else
		tmp = expm1(x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.65], N[(x * x), $MachinePrecision], N[(Exp[x] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6499999999999999

   1. Initial program 68.5%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Step-by-step derivation

      1. associate-+l- 68.7%

         \[\leadsto \color{blue}{e^{x} - \left(2 - e^{-x}\right)} \]

      2. sub-neg 68.7%

         \[\leadsto \color{blue}{e^{x} + \left(-\left(2 - e^{-x}\right)\right)} \]

      3. sub-neg 68.7%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(2 + \left(-e^{-x}\right)\right)}\right) \]

      4. +-commutative 68.7%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(\left(-e^{-x}\right) + 2\right)}\right) \]

      5. distribute-neg-in 68.7%

         \[\leadsto e^{x} + \color{blue}{\left(\left(-\left(-e^{-x}\right)\right) + \left(-2\right)\right)} \]

      6. remove-double-neg 68.7%

         \[\leadsto e^{x} + \left(\color{blue}{e^{-x}} + \left(-2\right)\right) \]

      7. metadata-eval 68.7%

         \[\leadsto e^{x} + \left(e^{-x} + \color{blue}{-2}\right) \]

   3. Simplified 68.7%

      \[\leadsto \color{blue}{e^{x} + \left(e^{-x} + -2\right)} \]

   4. Taylor expanded in x around 0 86.1%

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

      1. unpow2 86.1%

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

   6. Simplified 86.1%

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

   if 1.6499999999999999 < x

   1. Initial program 100.0%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{e^{x} - \left(2 - e^{-x}\right)} \]

      2. sub-neg 100.0%

         \[\leadsto \color{blue}{e^{x} + \left(-\left(2 - e^{-x}\right)\right)} \]

      3. sub-neg 100.0%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(2 + \left(-e^{-x}\right)\right)}\right) \]

      4. +-commutative 100.0%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(\left(-e^{-x}\right) + 2\right)}\right) \]

      5. distribute-neg-in 100.0%

         \[\leadsto e^{x} + \color{blue}{\left(\left(-\left(-e^{-x}\right)\right) + \left(-2\right)\right)} \]

      6. remove-double-neg 100.0%

         \[\leadsto e^{x} + \left(\color{blue}{e^{-x}} + \left(-2\right)\right) \]

      7. metadata-eval 100.0%

         \[\leadsto e^{x} + \left(e^{-x} + \color{blue}{-2}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x} + \left(e^{-x} + -2\right)} \]

   4. Taylor expanded in x around 0 98.8%

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

   5. Taylor expanded in x around inf 98.8%

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

      1. expm1-def 98.8%

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

   7. Simplified 98.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 6: 75.8% accurate, 68.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.4%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Step-by-step derivation

   1. associate-+l- 76.5%

      \[\leadsto \color{blue}{e^{x} - \left(2 - e^{-x}\right)} \]

   2. sub-neg 76.5%

      \[\leadsto \color{blue}{e^{x} + \left(-\left(2 - e^{-x}\right)\right)} \]

   3. sub-neg 76.5%

      \[\leadsto e^{x} + \left(-\color{blue}{\left(2 + \left(-e^{-x}\right)\right)}\right) \]

   4. +-commutative 76.5%

      \[\leadsto e^{x} + \left(-\color{blue}{\left(\left(-e^{-x}\right) + 2\right)}\right) \]

   5. distribute-neg-in 76.5%

      \[\leadsto e^{x} + \color{blue}{\left(\left(-\left(-e^{-x}\right)\right) + \left(-2\right)\right)} \]

   6. remove-double-neg 76.5%

      \[\leadsto e^{x} + \left(\color{blue}{e^{-x}} + \left(-2\right)\right) \]

   7. metadata-eval 76.5%

      \[\leadsto e^{x} + \left(e^{-x} + \color{blue}{-2}\right) \]

3. Simplified 76.5%

   \[\leadsto \color{blue}{e^{x} + \left(e^{-x} + -2\right)} \]

4. Taylor expanded in x around 0 77.1%

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

   1. unpow2 77.1%

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

6. Simplified 77.1%

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

7. Final simplification 77.1%

   \[\leadsto x \cdot x \]

Alternative 7: 26.3% accurate, 206.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 76.4%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Step-by-step derivation

   1. associate-+l- 76.5%

      \[\leadsto \color{blue}{e^{x} - \left(2 - e^{-x}\right)} \]

   2. sub-neg 76.5%

      \[\leadsto \color{blue}{e^{x} + \left(-\left(2 - e^{-x}\right)\right)} \]

   3. sub-neg 76.5%

      \[\leadsto e^{x} + \left(-\color{blue}{\left(2 + \left(-e^{-x}\right)\right)}\right) \]

   4. +-commutative 76.5%

      \[\leadsto e^{x} + \left(-\color{blue}{\left(\left(-e^{-x}\right) + 2\right)}\right) \]

   5. distribute-neg-in 76.5%

      \[\leadsto e^{x} + \color{blue}{\left(\left(-\left(-e^{-x}\right)\right) + \left(-2\right)\right)} \]

   6. remove-double-neg 76.5%

      \[\leadsto e^{x} + \left(\color{blue}{e^{-x}} + \left(-2\right)\right) \]

   7. metadata-eval 76.5%

      \[\leadsto e^{x} + \left(e^{-x} + \color{blue}{-2}\right) \]

3. Simplified 76.5%

   \[\leadsto \color{blue}{e^{x} + \left(e^{-x} + -2\right)} \]

5. Applied egg-rr 26.9%

   \[\leadsto \color{blue}{0} \]

6. Final simplification 26.9%

   \[\leadsto 0 \]

Developer target: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 4 \cdot {\sinh \left(\frac{x}{2}\right)}^{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 4.0 (pow (sinh (/ x 2.0)) 2.0)))

C

double code(double x) {
	return 4.0 * pow(sinh((x / 2.0)), 2.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 4.0d0 * (sinh((x / 2.0d0)) ** 2.0d0)
end function

Java

public static double code(double x) {
	return 4.0 * Math.pow(Math.sinh((x / 2.0)), 2.0);
}

Python

def code(x):
	return 4.0 * math.pow(math.sinh((x / 2.0)), 2.0)

Julia

function code(x)
	return Float64(4.0 * (sinh(Float64(x / 2.0)) ^ 2.0))
end

MATLAB

function tmp = code(x)
	tmp = 4.0 * (sinh((x / 2.0)) ^ 2.0);
end

Wolfram

code[x_] := N[(4.0 * N[Power[N[Sinh[N[(x / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023171 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

  :herbie-target
  (* 4.0 (pow (sinh (/ x 2.0)) 2.0))

  (+ (- (exp x) 2.0) (exp (- x))))