exp2 (problem 3.3.7)

Percentage Accurate: 76.9% → 99.9%

Time: 4.7s

Alternatives: 8

Speedup: 2.0×

• 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: 76.9% 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: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(e^{x} - 2\right) + t_0 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(0.002777777777777778, {x}^{6}, x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(t_0 + -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (+ (- (exp x) 2.0) t_0) 5e-7)
     (fma
      0.002777777777777778
      (pow x 6.0)
      (+ (* 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) <= 5e-7) {
		tmp = fma(0.002777777777777778, pow(x, 6.0), ((x * x) + (0.08333333333333333 * pow(x, 4.0))));
	} else {
		tmp = 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) <= 5e-7)
		tmp = fma(0.002777777777777778, (x ^ 6.0), Float64(Float64(x * x) + Float64(0.08333333333333333 * (x ^ 4.0))));
	else
		tmp = Float64(exp(x) + Float64(t_0 + -2.0));
	end
	return 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], 5e-7], N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision] + N[(N[(x * x), $MachinePrecision] + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $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))) < 4.99999999999999977e-7

   1. Initial program 51.6%

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

      1. associate-+l- 51.7%

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

      2. sub-neg 51.7%

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

      3. sub-neg 51.7%

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

      4. +-commutative 51.7%

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

      5. distribute-neg-in 51.7%

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

      6. remove-double-neg 51.7%

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

      7. metadata-eval 51.7%

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

   3. Simplified 51.7%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{0.002777777777777778 \cdot {x}^{6} + \left({x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)} \]
   5. Step-by-step derivation

      1. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.002777777777777778, {x}^{6}, {x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)} \]

      2. unpow2 100.0%

         \[\leadsto \mathsf{fma}\left(0.002777777777777778, {x}^{6}, \color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4}\right) \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.002777777777777778, {x}^{6}, x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)} \]

   if 4.99999999999999977e-7 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 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%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(0.002777777777777778, {x}^{6}, x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(e^{-x} + -2\right)\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(e^{x} - 2\right) + t_0 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(t_0 + -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (+ (- (exp x) 2.0) t_0) 5e-7)
     (+ (* 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) <= 5e-7) {
		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) <= 5d-7) 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) <= 5e-7) {
		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) <= 5e-7:
		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) <= 5e-7)
		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) <= 5e-7)
		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], 5e-7], 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))) < 4.99999999999999977e-7

   1. Initial program 51.6%

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

      1. associate-+l- 51.7%

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

      2. sub-neg 51.7%

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

      3. sub-neg 51.7%

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

      4. +-commutative 51.7%

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

      5. distribute-neg-in 51.7%

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

      6. remove-double-neg 51.7%

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

      7. metadata-eval 51.7%

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

   3. Simplified 51.7%

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

   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 4.99999999999999977e-7 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 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%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(e^{-x} + -2\right)\\ \end{array} \]

Alternative 3: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2:\\ \;\;\;\;\sqrt{{x}^{12} \cdot 7.71604938271605 \cdot 10^{-6}}\\ \mathbf{elif}\;x \leq 2.6:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -6.2)
   (sqrt (* (pow x 12.0) 7.71604938271605e-6))
   (if (<= x 2.6) (+ (* x x) (* 0.08333333333333333 (pow x 4.0))) (expm1 x))))

C

double code(double x) {
	double tmp;
	if (x <= -6.2) {
		tmp = sqrt((pow(x, 12.0) * 7.71604938271605e-6));
	} 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 <= -6.2) {
		tmp = Math.sqrt((Math.pow(x, 12.0) * 7.71604938271605e-6));
	} 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 <= -6.2:
		tmp = math.sqrt((math.pow(x, 12.0) * 7.71604938271605e-6))
	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 <= -6.2)
		tmp = sqrt(Float64((x ^ 12.0) * 7.71604938271605e-6));
	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, -6.2], N[Sqrt[N[(N[Power[x, 12.0], $MachinePrecision] * 7.71604938271605e-6), $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 < -6.20000000000000018

   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 85.6%

      \[\leadsto \color{blue}{0.002777777777777778 \cdot {x}^{6} + \left({x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)} \]
   5. Step-by-step derivation

      1. fma-def 85.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.002777777777777778, {x}^{6}, {x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)} \]

      2. unpow2 85.6%

         \[\leadsto \mathsf{fma}\left(0.002777777777777778, {x}^{6}, \color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4}\right) \]

   6. Simplified 85.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.002777777777777778, {x}^{6}, x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)} \]

   7. Taylor expanded in x around inf 85.6%

      \[\leadsto \color{blue}{0.002777777777777778 \cdot {x}^{6}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 85.6%

         \[\leadsto \color{blue}{\sqrt{0.002777777777777778 \cdot {x}^{6}} \cdot \sqrt{0.002777777777777778 \cdot {x}^{6}}} \]

      2. sqrt-unprod 93.3%

         \[\leadsto \color{blue}{\sqrt{\left(0.002777777777777778 \cdot {x}^{6}\right) \cdot \left(0.002777777777777778 \cdot {x}^{6}\right)}} \]

      3. *-commutative 93.3%

         \[\leadsto \sqrt{\color{blue}{\left({x}^{6} \cdot 0.002777777777777778\right)} \cdot \left(0.002777777777777778 \cdot {x}^{6}\right)} \]

      4. *-commutative 93.3%

         \[\leadsto \sqrt{\left({x}^{6} \cdot 0.002777777777777778\right) \cdot \color{blue}{\left({x}^{6} \cdot 0.002777777777777778\right)}} \]

      5. swap-sqr 93.3%

         \[\leadsto \sqrt{\color{blue}{\left({x}^{6} \cdot {x}^{6}\right) \cdot \left(0.002777777777777778 \cdot 0.002777777777777778\right)}} \]

      6. pow-prod-up 93.3%

         \[\leadsto \sqrt{\color{blue}{{x}^{\left(6 + 6\right)}} \cdot \left(0.002777777777777778 \cdot 0.002777777777777778\right)} \]

      7. metadata-eval 93.3%

         \[\leadsto \sqrt{{x}^{\color{blue}{12}} \cdot \left(0.002777777777777778 \cdot 0.002777777777777778\right)} \]

      8. metadata-eval 93.3%

         \[\leadsto \sqrt{{x}^{12} \cdot \color{blue}{7.71604938271605 \cdot 10^{-6}}} \]

   9. Applied egg-rr 93.3%

      \[\leadsto \color{blue}{\sqrt{{x}^{12} \cdot 7.71604938271605 \cdot 10^{-6}}} \]

   if -6.20000000000000018 < x < 2.60000000000000009

   1. Initial program 51.6%

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

      1. associate-+l- 51.7%

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

      2. sub-neg 51.7%

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

      3. sub-neg 51.7%

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

      4. +-commutative 51.7%

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

      5. distribute-neg-in 51.7%

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

      6. remove-double-neg 51.7%

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

      7. metadata-eval 51.7%

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

   3. Simplified 51.7%

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

   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 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 99.1%

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

   5. Taylor expanded in x around inf 99.1%

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

      1. expm1-def 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2:\\ \;\;\;\;\sqrt{{x}^{12} \cdot 7.71604938271605 \cdot 10^{-6}}\\ \mathbf{elif}\;x \leq 2.6:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 4: 95.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2:\\ \;\;\;\;0.002777777777777778 \cdot {x}^{6}\\ \mathbf{elif}\;x \leq 2.6:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -6.2)
   (* 0.002777777777777778 (pow x 6.0))
   (if (<= x 2.6) (+ (* x x) (* 0.08333333333333333 (pow x 4.0))) (expm1 x))))

C

double code(double x) {
	double tmp;
	if (x <= -6.2) {
		tmp = 0.002777777777777778 * pow(x, 6.0);
	} 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 <= -6.2) {
		tmp = 0.002777777777777778 * Math.pow(x, 6.0);
	} 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 <= -6.2:
		tmp = 0.002777777777777778 * math.pow(x, 6.0)
	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 <= -6.2)
		tmp = Float64(0.002777777777777778 * (x ^ 6.0));
	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, -6.2], N[(0.002777777777777778 * N[Power[x, 6.0], $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 < -6.20000000000000018

   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 85.6%

      \[\leadsto \color{blue}{0.002777777777777778 \cdot {x}^{6} + \left({x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)} \]
   5. Step-by-step derivation

      1. fma-def 85.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.002777777777777778, {x}^{6}, {x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)} \]

      2. unpow2 85.6%

         \[\leadsto \mathsf{fma}\left(0.002777777777777778, {x}^{6}, \color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4}\right) \]

   6. Simplified 85.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.002777777777777778, {x}^{6}, x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)} \]

   7. Taylor expanded in x around inf 85.6%

      \[\leadsto \color{blue}{0.002777777777777778 \cdot {x}^{6}} \]

   if -6.20000000000000018 < x < 2.60000000000000009

   1. Initial program 51.6%

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

      1. associate-+l- 51.7%

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

      2. sub-neg 51.7%

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

      3. sub-neg 51.7%

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

      4. +-commutative 51.7%

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

      5. distribute-neg-in 51.7%

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

      6. remove-double-neg 51.7%

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

      7. metadata-eval 51.7%

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

   3. Simplified 51.7%

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

   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 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 99.1%

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

   5. Taylor expanded in x around inf 99.1%

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

      1. expm1-def 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2:\\ \;\;\;\;0.002777777777777778 \cdot {x}^{6}\\ \mathbf{elif}\;x \leq 2.6:\\ \;\;\;\;x \cdot x + 0.08333333333333333 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 5: 95.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4:\\ \;\;\;\;0.002777777777777778 \cdot {x}^{6}\\ \mathbf{elif}\;x \leq 1.65:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.4)
   (* 0.002777777777777778 (pow x 6.0))
   (if (<= x 1.65) (* x x) (expm1 x))))

C

double code(double x) {
	double tmp;
	if (x <= -4.4) {
		tmp = 0.002777777777777778 * pow(x, 6.0);
	} 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 <= -4.4) {
		tmp = 0.002777777777777778 * Math.pow(x, 6.0);
	} else if (x <= 1.65) {
		tmp = x * x;
	} else {
		tmp = Math.expm1(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.4:
		tmp = 0.002777777777777778 * math.pow(x, 6.0)
	elif x <= 1.65:
		tmp = x * x
	else:
		tmp = math.expm1(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.4)
		tmp = Float64(0.002777777777777778 * (x ^ 6.0));
	elseif (x <= 1.65)
		tmp = Float64(x * x);
	else
		tmp = expm1(x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -4.4], N[(0.002777777777777778 * N[Power[x, 6.0], $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 < -4.4000000000000004

   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 85.6%

      \[\leadsto \color{blue}{0.002777777777777778 \cdot {x}^{6} + \left({x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)} \]
   5. Step-by-step derivation

      1. fma-def 85.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.002777777777777778, {x}^{6}, {x}^{2} + 0.08333333333333333 \cdot {x}^{4}\right)} \]

      2. unpow2 85.6%

         \[\leadsto \mathsf{fma}\left(0.002777777777777778, {x}^{6}, \color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4}\right) \]

   6. Simplified 85.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.002777777777777778, {x}^{6}, x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)} \]

   7. Taylor expanded in x around inf 85.6%

      \[\leadsto \color{blue}{0.002777777777777778 \cdot {x}^{6}} \]

   if -4.4000000000000004 < x < 1.6499999999999999

   1. Initial program 51.6%

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

      1. associate-+l- 51.7%

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

      2. sub-neg 51.7%

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

      3. sub-neg 51.7%

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

      4. +-commutative 51.7%

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

      5. distribute-neg-in 51.7%

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

      6. remove-double-neg 51.7%

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

      7. metadata-eval 51.7%

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

   3. Simplified 51.7%

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

   4. Taylor expanded in x around 0 99.5%

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

      1. unpow2 99.5%

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

   6. Simplified 99.5%

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

   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 99.1%

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

   5. Taylor expanded in x around inf 99.1%

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

      1. expm1-def 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4:\\ \;\;\;\;0.002777777777777778 \cdot {x}^{6}\\ \mathbf{elif}\;x \leq 1.65:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 6: 87.8% 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 69.9%

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

      1. associate-+l- 69.9%

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

      2. sub-neg 69.9%

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

      3. sub-neg 69.9%

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

      4. +-commutative 69.9%

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

      5. distribute-neg-in 69.9%

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

      6. remove-double-neg 69.9%

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

      7. metadata-eval 69.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in x around 0 80.4%

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

      1. unpow2 80.4%

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

   6. Simplified 80.4%

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

   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 99.1%

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

   5. Taylor expanded in x around inf 99.1%

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

      1. expm1-def 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 85.1%

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

Alternative 7: 76.4% 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 77.5%

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

   1. associate-+l- 77.5%

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

   2. sub-neg 77.5%

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

   3. sub-neg 77.5%

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

   4. +-commutative 77.5%

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

   5. distribute-neg-in 77.5%

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

   6. remove-double-neg 77.5%

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

   7. metadata-eval 77.5%

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

3. Simplified 77.5%

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

4. Taylor expanded in x around 0 71.4%

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

   1. unpow2 71.4%

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

6. Simplified 71.4%

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

7. Final simplification 71.4%

   \[\leadsto x \cdot x \]

Alternative 8: 26.8% 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 77.5%

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

   1. associate-+l- 77.5%

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

   2. sub-neg 77.5%

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

   3. sub-neg 77.5%

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

   4. +-commutative 77.5%

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

   5. distribute-neg-in 77.5%

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

   6. remove-double-neg 77.5%

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

   7. metadata-eval 77.5%

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

3. Simplified 77.5%

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

5. Applied egg-rr 24.4%

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

6. Final simplification 24.4%

   \[\leadsto 0 \]

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 2023195 
(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))))