exp2 (problem 3.3.7)

Percentage Accurate: 76.5% → 100.0%

Time: 13.3s

Alternatives: 12

Speedup: 2.0×

• 49.7% 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: 76.5% 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: 100.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(e^{x} - 2\right) + t_0 \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, {x}^{8}, \left(0.08333333333333333 \cdot {x}^{4} + 0.002777777777777778 \cdot {x}^{6}\right) + x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(t_0 + -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (+ (- (exp x) 2.0) t_0) 0.001)
     (fma
      4.96031746031746e-5
      (pow x 8.0)
      (+
       (+
        (* 0.08333333333333333 (pow x 4.0))
        (* 0.002777777777777778 (pow x 6.0)))
       (* x x)))
     (+ (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) <= 0.001) {
		tmp = fma(4.96031746031746e-5, pow(x, 8.0), (((0.08333333333333333 * pow(x, 4.0)) + (0.002777777777777778 * pow(x, 6.0))) + (x * x)));
	} 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) <= 0.001)
		tmp = fma(4.96031746031746e-5, (x ^ 8.0), Float64(Float64(Float64(0.08333333333333333 * (x ^ 4.0)) + Float64(0.002777777777777778 * (x ^ 6.0))) + Float64(x * x)));
	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], 0.001], N[(4.96031746031746e-5 * N[Power[x, 8.0], $MachinePrecision] + N[(N[(N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * x), $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))) < 1e-3

   1. Initial program 52.5%

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

      1. associate-+l- 52.6%

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

      2. sub-neg 52.6%

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

      3. sub-neg 52.6%

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

      4. +-commutative 52.6%

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

      5. distribute-neg-in 52.6%

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

      6. remove-double-neg 52.6%

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

      7. metadata-eval 52.6%

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

   3. Simplified 52.6%

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

   4. Taylor expanded in x around 0 99.9%

      \[\leadsto \color{blue}{4.96031746031746 \cdot 10^{-5} \cdot {x}^{8} + \left(0.002777777777777778 \cdot {x}^{6} + \left(0.08333333333333333 \cdot {x}^{4} + {x}^{2}\right)\right)} \]
   5. Step-by-step derivation

      1. fma-def 99.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, {x}^{8}, 0.002777777777777778 \cdot {x}^{6} + \left(0.08333333333333333 \cdot {x}^{4} + {x}^{2}\right)\right)} \]

      2. fma-def 99.9%

         \[\leadsto \mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, {x}^{8}, \color{blue}{\mathsf{fma}\left(0.002777777777777778, {x}^{6}, 0.08333333333333333 \cdot {x}^{4} + {x}^{2}\right)}\right) \]

      3. +-commutative 99.9%

         \[\leadsto \mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, {x}^{8}, \mathsf{fma}\left(0.002777777777777778, {x}^{6}, \color{blue}{{x}^{2} + 0.08333333333333333 \cdot {x}^{4}}\right)\right) \]

      4. unpow2 99.9%

         \[\leadsto \mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, {x}^{8}, \mathsf{fma}\left(0.002777777777777778, {x}^{6}, \color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4}\right)\right) \]

      5. fma-def 100.0%

         \[\leadsto \mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, {x}^{8}, \mathsf{fma}\left(0.002777777777777778, {x}^{6}, \color{blue}{\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)}\right)\right) \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, {x}^{8}, \mathsf{fma}\left(0.002777777777777778, {x}^{6}, \mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. fma-udef 100.0%

         \[\leadsto \mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, {x}^{8}, \color{blue}{0.002777777777777778 \cdot {x}^{6} + \mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)}\right) \]

      2. fma-udef 99.9%

         \[\leadsto \mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, {x}^{8}, 0.002777777777777778 \cdot {x}^{6} + \color{blue}{\left(x \cdot x + 0.08333333333333333 \cdot {x}^{4}\right)}\right) \]

      3. unpow2 99.9%

         \[\leadsto \mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, {x}^{8}, 0.002777777777777778 \cdot {x}^{6} + \left(\color{blue}{{x}^{2}} + 0.08333333333333333 \cdot {x}^{4}\right)\right) \]

      4. +-commutative 99.9%

         \[\leadsto \mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, {x}^{8}, 0.002777777777777778 \cdot {x}^{6} + \color{blue}{\left(0.08333333333333333 \cdot {x}^{4} + {x}^{2}\right)}\right) \]

      5. associate-+r+ 100.0%

         \[\leadsto \mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, {x}^{8}, \color{blue}{\left(0.002777777777777778 \cdot {x}^{6} + 0.08333333333333333 \cdot {x}^{4}\right) + {x}^{2}}\right) \]

      6. fma-def 100.0%

         \[\leadsto \mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, {x}^{8}, \color{blue}{\mathsf{fma}\left(0.002777777777777778, {x}^{6}, 0.08333333333333333 \cdot {x}^{4}\right)} + {x}^{2}\right) \]

      7. unpow2 100.0%

         \[\leadsto \mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, {x}^{8}, \mathsf{fma}\left(0.002777777777777778, {x}^{6}, 0.08333333333333333 \cdot {x}^{4}\right) + \color{blue}{x \cdot x}\right) \]

   8. Applied egg-rr 100.0%

      \[\leadsto \mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, {x}^{8}, \color{blue}{\mathsf{fma}\left(0.002777777777777778, {x}^{6}, 0.08333333333333333 \cdot {x}^{4}\right) + x \cdot x}\right) \]
   9. Step-by-step derivation

      1. fma-udef 100.0%

         \[\leadsto \mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, {x}^{8}, \color{blue}{\left(0.002777777777777778 \cdot {x}^{6} + 0.08333333333333333 \cdot {x}^{4}\right)} + x \cdot x\right) \]

      2. +-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, {x}^{8}, \color{blue}{\left(0.08333333333333333 \cdot {x}^{4} + 0.002777777777777778 \cdot {x}^{6}\right)} + x \cdot x\right) \]

   10. Applied egg-rr 100.0%

       \[\leadsto \mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, {x}^{8}, \color{blue}{\left(0.08333333333333333 \cdot {x}^{4} + 0.002777777777777778 \cdot {x}^{6}\right)} + x \cdot x\right) \]

   if 1e-3 < (+.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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 0.001:\\ \;\;\;\;\mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, {x}^{8}, \left(0.08333333333333333 \cdot {x}^{4} + 0.002777777777777778 \cdot {x}^{6}\right) + x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(e^{-x} + -2\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(e^{x} - 2\right) + t_0 \leq 0.0009:\\ \;\;\;\;0.002777777777777778 \cdot {x}^{6} + \mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \log \left(1 + \mathsf{expm1}\left(2 - e^{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (+ (- (exp x) 2.0) t_0) 0.0009)
     (+
      (* 0.002777777777777778 (pow x 6.0))
      (fma x x (* 0.08333333333333333 (pow x 4.0))))
     (- t_0 (log (+ 1.0 (expm1 (- 2.0 (exp x)))))))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (((exp(x) - 2.0) + t_0) <= 0.0009) {
		tmp = (0.002777777777777778 * pow(x, 6.0)) + fma(x, x, (0.08333333333333333 * pow(x, 4.0)));
	} else {
		tmp = t_0 - log((1.0 + expm1((2.0 - exp(x)))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(Float64(exp(x) - 2.0) + t_0) <= 0.0009)
		tmp = Float64(Float64(0.002777777777777778 * (x ^ 6.0)) + fma(x, x, Float64(0.08333333333333333 * (x ^ 4.0))));
	else
		tmp = Float64(t_0 - log(Float64(1.0 + expm1(Float64(2.0 - exp(x))))));
	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], 0.0009], N[(N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(x * x + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[Log[N[(1.0 + N[(Exp[N[(2.0 - N[Exp[x], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 8.9999999999999998e-4

   1. Initial program 52.3%

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

      1. associate-+l- 52.3%

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

      2. sub-neg 52.3%

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

      3. sub-neg 52.3%

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

      4. +-commutative 52.3%

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

      5. distribute-neg-in 52.3%

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

      6. remove-double-neg 52.3%

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

      7. metadata-eval 52.3%

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

   3. Simplified 52.3%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. fma-udef 99.8%

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

   6. Applied egg-rr 99.8%

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

   if 8.9999999999999998e-4 < (+.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. +-commutative 99.9%

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

      2. associate-+r- 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r- 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+l- 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{e^{-x} - \left(2 - e^{x}\right)} \]
   4. Step-by-step derivation

      1. log1p-expm1-u 99.9%

         \[\leadsto e^{-x} - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(2 - e^{x}\right)\right)} \]

      2. log1p-udef 99.9%

         \[\leadsto e^{-x} - \color{blue}{\log \left(1 + \mathsf{expm1}\left(2 - e^{x}\right)\right)} \]

   5. Applied egg-rr 99.9%

      \[\leadsto e^{-x} - \color{blue}{\log \left(1 + \mathsf{expm1}\left(2 - e^{x}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 0.0009:\\ \;\;\;\;0.002777777777777778 \cdot {x}^{6} + \mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-x} - \log \left(1 + \mathsf{expm1}\left(2 - e^{x}\right)\right)\\ \end{array} \]

Alternative 3: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(e^{x} - 2\right) + t_0 \leq 0.001:\\ \;\;\;\;0.002777777777777778 \cdot {x}^{6} + \mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(t_0 + -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (+ (- (exp x) 2.0) t_0) 0.001)
     (+
      (* 0.002777777777777778 (pow x 6.0))
      (fma 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) <= 0.001) {
		tmp = (0.002777777777777778 * pow(x, 6.0)) + fma(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) <= 0.001)
		tmp = Float64(Float64(0.002777777777777778 * (x ^ 6.0)) + fma(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], 0.001], N[(N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(x * x + 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))) < 1e-3

   1. Initial program 52.5%

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

      1. associate-+l- 52.6%

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

      2. sub-neg 52.6%

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

      3. sub-neg 52.6%

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

      4. +-commutative 52.6%

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

      5. distribute-neg-in 52.6%

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

      6. remove-double-neg 52.6%

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

      7. metadata-eval 52.6%

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

   3. Simplified 52.6%

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

   4. Taylor expanded in x around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. fma-udef 99.6%

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

   6. Applied egg-rr 99.6%

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

   if 1e-3 < (+.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 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 0.001:\\ \;\;\;\;0.002777777777777778 \cdot {x}^{6} + \mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(e^{-x} + -2\right)\\ \end{array} \]

Alternative 4: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(e^{x} - 2\right) + t_0 \leq 0.001:\\ \;\;\;\;0.002777777777777778 \cdot {x}^{6} + \left(0.08333333333333333 \cdot {x}^{4} + x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(t_0 + -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (+ (- (exp x) 2.0) t_0) 0.001)
     (+
      (* 0.002777777777777778 (pow x 6.0))
      (+ (* 0.08333333333333333 (pow x 4.0)) (* x x)))
     (+ (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) <= 0.001) {
		tmp = (0.002777777777777778 * pow(x, 6.0)) + ((0.08333333333333333 * pow(x, 4.0)) + (x * x));
	} 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) <= 0.001d0) then
        tmp = (0.002777777777777778d0 * (x ** 6.0d0)) + ((0.08333333333333333d0 * (x ** 4.0d0)) + (x * x))
    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) <= 0.001) {
		tmp = (0.002777777777777778 * Math.pow(x, 6.0)) + ((0.08333333333333333 * Math.pow(x, 4.0)) + (x * x));
	} 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) <= 0.001:
		tmp = (0.002777777777777778 * math.pow(x, 6.0)) + ((0.08333333333333333 * math.pow(x, 4.0)) + (x * x))
	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) <= 0.001)
		tmp = Float64(Float64(0.002777777777777778 * (x ^ 6.0)) + Float64(Float64(0.08333333333333333 * (x ^ 4.0)) + Float64(x * x)));
	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) <= 0.001)
		tmp = (0.002777777777777778 * (x ^ 6.0)) + ((0.08333333333333333 * (x ^ 4.0)) + (x * x));
	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], 0.001], N[(N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x * x), $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))) < 1e-3

   1. Initial program 52.5%

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

      1. associate-+l- 52.6%

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

      2. sub-neg 52.6%

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

      3. sub-neg 52.6%

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

      4. +-commutative 52.6%

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

      5. distribute-neg-in 52.6%

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

      6. remove-double-neg 52.6%

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

      7. metadata-eval 52.6%

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

   3. Simplified 52.6%

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

   4. Taylor expanded in x around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. unpow2 99.6%

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

      3. fma-udef 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. fma-udef 99.6%

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

   8. Applied egg-rr 99.6%

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

   if 1e-3 < (+.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 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 0.001:\\ \;\;\;\;0.002777777777777778 \cdot {x}^{6} + \left(0.08333333333333333 \cdot {x}^{4} + x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(e^{-x} + -2\right)\\ \end{array} \]

Alternative 5: 99.9% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (exp x) 2.0) (exp (- x)))))
   (if (<= t_0 2e-5) (+ (* 0.08333333333333333 (pow x 4.0)) (* x x)) t_0)))

C

double code(double x) {
	double t_0 = (exp(x) - 2.0) + exp(-x);
	double tmp;
	if (t_0 <= 2e-5) {
		tmp = (0.08333333333333333 * pow(x, 4.0)) + (x * x);
	} else {
		tmp = t_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) - 2.0d0) + exp(-x)
    if (t_0 <= 2d-5) then
        tmp = (0.08333333333333333d0 * (x ** 4.0d0)) + (x * x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (Math.exp(x) - 2.0) + Math.exp(-x);
	double tmp;
	if (t_0 <= 2e-5) {
		tmp = (0.08333333333333333 * Math.pow(x, 4.0)) + (x * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (math.exp(x) - 2.0) + math.exp(-x)
	tmp = 0
	if t_0 <= 2e-5:
		tmp = (0.08333333333333333 * math.pow(x, 4.0)) + (x * x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(exp(x) - 2.0) + exp(Float64(-x)))
	tmp = 0.0
	if (t_0 <= 2e-5)
		tmp = Float64(Float64(0.08333333333333333 * (x ^ 4.0)) + Float64(x * x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (exp(x) - 2.0) + exp(-x);
	tmp = 0.0;
	if (t_0 <= 2e-5)
		tmp = (0.08333333333333333 * (x ^ 4.0)) + (x * x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-5], N[(N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 2.00000000000000016e-5

   1. Initial program 51.4%

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

      1. associate-+l- 51.5%

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

      2. sub-neg 51.5%

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

      3. sub-neg 51.5%

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

      4. +-commutative 51.5%

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

      5. distribute-neg-in 51.5%

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

      6. remove-double-neg 51.5%

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

      7. metadata-eval 51.5%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in x around 0 99.8%

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

      1. fma-def 99.8%

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

      2. unpow2 99.8%

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

   6. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.08333333333333333, {x}^{4}, x \cdot x\right)} \]
   7. Step-by-step derivation

      1. fma-udef 99.8%

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

      2. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   if 2.00000000000000016e-5 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 99.5%

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

4. Final simplification 99.7%

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

Alternative 6: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0046:\\ \;\;\;\;0.08333333333333333 \cdot {x}^{4} + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(e^{-x} + -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0046)
   (+ (* 0.08333333333333333 (pow x 4.0)) (* x x))
   (+ (exp x) (+ (exp (- x)) -2.0))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0046) {
		tmp = (0.08333333333333333 * pow(x, 4.0)) + (x * x);
	} else {
		tmp = exp(x) + (exp(-x) + -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0046d0) then
        tmp = (0.08333333333333333d0 * (x ** 4.0d0)) + (x * x)
    else
        tmp = exp(x) + (exp(-x) + (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.0046) {
		tmp = (0.08333333333333333 * Math.pow(x, 4.0)) + (x * x);
	} else {
		tmp = Math.exp(x) + (Math.exp(-x) + -2.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0046:
		tmp = (0.08333333333333333 * math.pow(x, 4.0)) + (x * x)
	else:
		tmp = math.exp(x) + (math.exp(-x) + -2.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0046)
		tmp = Float64(Float64(0.08333333333333333 * (x ^ 4.0)) + Float64(x * x));
	else
		tmp = Float64(exp(x) + Float64(exp(Float64(-x)) + -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0046)
		tmp = (0.08333333333333333 * (x ^ 4.0)) + (x * x);
	else
		tmp = exp(x) + (exp(-x) + -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0046], N[(N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[Exp[x], $MachinePrecision] + N[(N[Exp[(-x)], $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0045999999999999999

   1. Initial program 69.3%

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

      1. associate-+l- 69.3%

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

      2. sub-neg 69.3%

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

      3. sub-neg 69.3%

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

      4. +-commutative 69.3%

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

      5. distribute-neg-in 69.3%

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

      6. remove-double-neg 69.3%

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

      7. metadata-eval 69.3%

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

   3. Simplified 69.3%

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

   4. Taylor expanded in x around 0 88.1%

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

      1. fma-def 88.1%

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

      2. unpow2 88.1%

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

   6. Simplified 88.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.08333333333333333, {x}^{4}, x \cdot x\right)} \]
   7. Step-by-step derivation

      1. fma-udef 88.1%

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

      2. +-commutative 88.1%

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

   8. Applied egg-rr 88.1%

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

   if 0.0045999999999999999 < x

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

      2. sub-neg 99.6%

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

      3. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. distribute-neg-in 99.6%

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

      6. remove-double-neg 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0046:\\ \;\;\;\;0.08333333333333333 \cdot {x}^{4} + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(e^{-x} + -2\right)\\ \end{array} \]

Alternative 7: 94.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0052:\\ \;\;\;\;0.08333333333333333 \cdot {x}^{4} + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;-2 + 2 \cdot \cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0052)
   (+ (* 0.08333333333333333 (pow x 4.0)) (* x x))
   (+ -2.0 (* 2.0 (cosh x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0052) {
		tmp = (0.08333333333333333 * pow(x, 4.0)) + (x * x);
	} else {
		tmp = -2.0 + (2.0 * cosh(x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0052d0) then
        tmp = (0.08333333333333333d0 * (x ** 4.0d0)) + (x * x)
    else
        tmp = (-2.0d0) + (2.0d0 * cosh(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.0052) {
		tmp = (0.08333333333333333 * Math.pow(x, 4.0)) + (x * x);
	} else {
		tmp = -2.0 + (2.0 * Math.cosh(x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0052:
		tmp = (0.08333333333333333 * math.pow(x, 4.0)) + (x * x)
	else:
		tmp = -2.0 + (2.0 * math.cosh(x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0052)
		tmp = Float64(Float64(0.08333333333333333 * (x ^ 4.0)) + Float64(x * x));
	else
		tmp = Float64(-2.0 + Float64(2.0 * cosh(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0052)
		tmp = (0.08333333333333333 * (x ^ 4.0)) + (x * x);
	else
		tmp = -2.0 + (2.0 * cosh(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0052], N[(N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision], N[(-2.0 + N[(2.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0051999999999999998

   1. Initial program 69.3%

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

      1. associate-+l- 69.3%

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

      2. sub-neg 69.3%

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

      3. sub-neg 69.3%

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

      4. +-commutative 69.3%

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

      5. distribute-neg-in 69.3%

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

      6. remove-double-neg 69.3%

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

      7. metadata-eval 69.3%

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

   3. Simplified 69.3%

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

   4. Taylor expanded in x around 0 88.1%

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

      1. fma-def 88.1%

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

      2. unpow2 88.1%

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

   6. Simplified 88.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.08333333333333333, {x}^{4}, x \cdot x\right)} \]
   7. Step-by-step derivation

      1. fma-udef 88.1%

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

      2. +-commutative 88.1%

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

   8. Applied egg-rr 88.1%

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

   if 0.0051999999999999998 < x

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. associate-+r- 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r- 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+l- 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{e^{-x} - \left(2 - e^{x}\right)} \]
   4. Step-by-step derivation

      1. associate--r- 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

      4. +-commutative 99.6%

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

      5. associate-+r+ 99.6%

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

      6. cosh-undef 99.6%

         \[\leadsto \color{blue}{2 \cdot \cosh x} + -2 \]

   5. Applied egg-rr 99.6%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0052:\\ \;\;\;\;0.08333333333333333 \cdot {x}^{4} + x \cdot x\\ \mathbf{else}:\\ \;\;\;\;-2 + 2 \cdot \cosh x\\ \end{array} \]

Alternative 8: 88.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.000185:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;-2 + 2 \cdot \cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.000185) (* x x) (+ -2.0 (* 2.0 (cosh x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.000185) {
		tmp = x * x;
	} else {
		tmp = -2.0 + (2.0 * cosh(x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.000185d0) then
        tmp = x * x
    else
        tmp = (-2.0d0) + (2.0d0 * cosh(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.000185) {
		tmp = x * x;
	} else {
		tmp = -2.0 + (2.0 * Math.cosh(x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.000185:
		tmp = x * x
	else:
		tmp = -2.0 + (2.0 * math.cosh(x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.000185)
		tmp = Float64(x * x);
	else
		tmp = Float64(-2.0 + Float64(2.0 * cosh(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.000185)
		tmp = x * x;
	else
		tmp = -2.0 + (2.0 * cosh(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.000185], N[(x * x), $MachinePrecision], N[(-2.0 + N[(2.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.85e-4

   1. Initial program 69.3%

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

      1. associate-+l- 69.3%

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

      2. sub-neg 69.3%

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

      3. sub-neg 69.3%

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

      4. +-commutative 69.3%

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

      5. distribute-neg-in 69.3%

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

      6. remove-double-neg 69.3%

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

      7. metadata-eval 69.3%

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

   3. Simplified 69.3%

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

   4. Taylor expanded in x around 0 81.2%

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

      1. unpow2 81.2%

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

   6. Simplified 81.2%

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

   if 1.85e-4 < x

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. associate-+r- 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+r- 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+l- 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{e^{-x} - \left(2 - e^{x}\right)} \]
   4. Step-by-step derivation

      1. associate--r- 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

      4. +-commutative 99.6%

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

      5. associate-+r+ 99.6%

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

      6. cosh-undef 99.6%

         \[\leadsto \color{blue}{2 \cdot \cosh x} + -2 \]

   5. Applied egg-rr 99.6%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000185:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;-2 + 2 \cdot \cosh x\\ \end{array} \]

Alternative 9: 87.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.7) (* x x) (expm1 x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.7) {
		tmp = x * x;
	} else {
		tmp = expm1(x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.7) {
		tmp = x * x;
	} else {
		tmp = Math.expm1(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.7:
		tmp = x * x
	else:
		tmp = math.expm1(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.7)
		tmp = Float64(x * x);
	else
		tmp = expm1(x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.7], N[(x * x), $MachinePrecision], N[(Exp[x] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.69999999999999996

   1. Initial program 69.7%

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

      1. associate-+l- 69.7%

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

      2. sub-neg 69.7%

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

      3. sub-neg 69.7%

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

      4. +-commutative 69.7%

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

      5. distribute-neg-in 69.7%

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

      6. remove-double-neg 69.7%

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

      7. metadata-eval 69.7%

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

   3. Simplified 69.7%

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

   4. Taylor expanded in x around 0 80.5%

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

      1. unpow2 80.5%

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

   6. Simplified 80.5%

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

   if 1.69999999999999996 < 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 0 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. expm1-def 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 10: 76.0% 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 78.3%

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

   1. associate-+l- 78.3%

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

   2. sub-neg 78.3%

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

   3. sub-neg 78.3%

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

   4. +-commutative 78.3%

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

   5. distribute-neg-in 78.3%

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

   6. remove-double-neg 78.3%

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

   7. metadata-eval 78.3%

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

3. Simplified 78.3%

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

4. Taylor expanded in x around 0 72.3%

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

   1. unpow2 72.3%

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

6. Simplified 72.3%

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

7. Final simplification 72.3%

   \[\leadsto x \cdot x \]

Alternative 11: 3.7% accurate, 206.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

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

Wolfram

code[x_] := 2.0

Derivation

1. Initial program 78.3%

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

   1. associate-+l- 78.3%

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

   2. sub-neg 78.3%

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

   3. sub-neg 78.3%

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

   4. +-commutative 78.3%

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

   5. distribute-neg-in 78.3%

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

   6. remove-double-neg 78.3%

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

   7. metadata-eval 78.3%

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

3. Simplified 78.3%

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

   1. +-commutative 78.3%

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

   2. metadata-eval 78.3%

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

   3. sub-neg 78.3%

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

   4. associate--r- 78.3%

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

   5. add-sqr-sqrt 38.0%

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

   6. sqrt-unprod 77.4%

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

   7. sqr-neg 77.4%

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

   8. sqrt-unprod 39.3%

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

   9. add-sqr-sqrt 51.2%

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

5. Applied egg-rr 51.2%

   \[\leadsto \color{blue}{e^{x} - \left(2 - e^{x}\right)} \]
6. Step-by-step derivation

   1. associate--r- 51.2%

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

   2. sub-neg 51.2%

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

   3. metadata-eval 51.2%

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

   4. +-commutative 51.2%

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

   5. rem-square-sqrt 28.5%

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

   6. fabs-sqr 28.5%

      \[\leadsto e^{x} + \color{blue}{\left|\sqrt{e^{x} + -2} \cdot \sqrt{e^{x} + -2}\right|} \]

   7. rem-square-sqrt 31.5%

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

   8. metadata-eval 31.5%

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

   9. sub-neg 31.5%

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

   10. fabs-sub 31.5%

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

   11. rem-square-sqrt 3.0%

       \[\leadsto e^{x} + \left|\color{blue}{\sqrt{2 - e^{x}} \cdot \sqrt{2 - e^{x}}}\right| \]

   12. fabs-sqr 3.0%

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

   13. rem-square-sqrt 3.0%

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

   14. remove-double-neg 3.0%

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

   15. remove-double-neg 3.0%

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

   16. distribute-neg-out 3.0%

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

   17. +-commutative 3.0%

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

   18. neg-sub0 3.0%

       \[\leadsto -\left(\color{blue}{\left(0 - \left(2 - e^{x}\right)\right)} + \left(-e^{x}\right)\right) \]

   19. associate--r- 3.0%

       \[\leadsto -\left(\color{blue}{\left(\left(0 - 2\right) + e^{x}\right)} + \left(-e^{x}\right)\right) \]

   20. metadata-eval 3.0%

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

7. Simplified 3.9%

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

8. Final simplification 3.9%

   \[\leadsto 2 \]

Alternative 12: 4.4% accurate, 206.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 78.3%

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

   1. associate-+l- 78.3%

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

   2. sub-neg 78.3%

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

   3. sub-neg 78.3%

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

   4. +-commutative 78.3%

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

   5. distribute-neg-in 78.3%

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

   6. remove-double-neg 78.3%

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

   7. metadata-eval 78.3%

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

3. Simplified 78.3%

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

4. Taylor expanded in x around 0 51.2%

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

5. Taylor expanded in x around 0 4.3%

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

6. Final simplification 4.3%

   \[\leadsto x \]

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 2023280 
(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))))