exp2 (problem 3.3.7)

Percentage Accurate: 54.2% → 100.0%

Time: 14.1s

Alternatives: 13

Speedup: 2.0×

Specification

\[\left|x\right| \leq 710\]

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: 54.2% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(e^{x} - 2\right) + t\_0 \leq 0.01:\\ \;\;\;\;{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + {x}^{2} \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(t\_0 + \left(-1 + \mathsf{expm1}\left(x\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (+ (- (exp x) 2.0) t_0) 0.01)
     (*
      (pow x 2.0)
      (+
       1.0
       (*
        (pow x 2.0)
        (+
         0.08333333333333333
         (*
          (pow x 2.0)
          (+ 0.002777777777777778 (* (pow x 2.0) 4.96031746031746e-5)))))))
     (exp (log (+ t_0 (+ -1.0 (expm1 x))))))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (((exp(x) - 2.0) + t_0) <= 0.01) {
		tmp = pow(x, 2.0) * (1.0 + (pow(x, 2.0) * (0.08333333333333333 + (pow(x, 2.0) * (0.002777777777777778 + (pow(x, 2.0) * 4.96031746031746e-5))))));
	} else {
		tmp = exp(log((t_0 + (-1.0 + expm1(x)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (((Math.exp(x) - 2.0) + t_0) <= 0.01) {
		tmp = Math.pow(x, 2.0) * (1.0 + (Math.pow(x, 2.0) * (0.08333333333333333 + (Math.pow(x, 2.0) * (0.002777777777777778 + (Math.pow(x, 2.0) * 4.96031746031746e-5))))));
	} else {
		tmp = Math.exp(Math.log((t_0 + (-1.0 + Math.expm1(x)))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.exp(-x)
	tmp = 0
	if ((math.exp(x) - 2.0) + t_0) <= 0.01:
		tmp = math.pow(x, 2.0) * (1.0 + (math.pow(x, 2.0) * (0.08333333333333333 + (math.pow(x, 2.0) * (0.002777777777777778 + (math.pow(x, 2.0) * 4.96031746031746e-5))))))
	else:
		tmp = math.exp(math.log((t_0 + (-1.0 + math.expm1(x)))))
	return tmp

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(Float64(exp(x) - 2.0) + t_0) <= 0.01)
		tmp = Float64((x ^ 2.0) * Float64(1.0 + Float64((x ^ 2.0) * Float64(0.08333333333333333 + Float64((x ^ 2.0) * Float64(0.002777777777777778 + Float64((x ^ 2.0) * 4.96031746031746e-5)))))));
	else
		tmp = exp(log(Float64(t_0 + Float64(-1.0 + expm1(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], 0.01], N[(N[Power[x, 2.0], $MachinePrecision] * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.08333333333333333 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.002777777777777778 + N[(N[Power[x, 2.0], $MachinePrecision] * 4.96031746031746e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[Log[N[(t$95$0 + N[(-1.0 + N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x))) < 0.0100000000000000002

   1. Initial program 56.0%

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

      1. associate-+l- 56.0%

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

      2. sub-neg 56.0%

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

      3. sub-neg 56.0%

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

      4. distribute-neg-in 56.0%

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

      5. remove-double-neg 56.0%

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

      6. +-commutative 56.0%

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

      7. metadata-eval 56.0%

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

   3. Simplified 56.0%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + 4.96031746031746 \cdot 10^{-5} \cdot {x}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + \color{blue}{{x}^{2} \cdot 4.96031746031746 \cdot 10^{-5}}\right)\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + {x}^{2} \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right)} \]

   if 0.0100000000000000002 < (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x)))

   1. Initial program 98.4%

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

      1. associate-+l- 98.2%

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

      2. sub-neg 98.2%

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

      3. sub-neg 98.2%

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

      4. distribute-neg-in 98.2%

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

      5. remove-double-neg 98.2%

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

      6. +-commutative 98.2%

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

      7. metadata-eval 98.2%

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

   3. Simplified 98.2%

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

      1. +-commutative 98.2%

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

      2. associate-+r+ 98.4%

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

      3. metadata-eval 98.4%

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

      4. sub-neg 98.4%

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

      5. add-exp-log 97.2%

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

      6. +-commutative 97.2%

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

      7. sub-neg 97.2%

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

      8. metadata-eval 97.2%

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

      9. associate-+r+ 97.0%

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

      10. +-commutative 97.0%

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

      11. +-commutative 97.0%

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

      12. cosh-undef 97.0%

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

   6. Applied egg-rr 97.0%

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

   7. Taylor expanded in x around inf 97.0%

      \[\leadsto e^{\color{blue}{\log \left(\left(e^{x} + \frac{1}{e^{x}}\right) - 2\right)}} \]
   8. Step-by-step derivation

      1. sub-neg 97.0%

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

      2. +-commutative 97.0%

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

      3. metadata-eval 97.0%

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

      4. associate-+l+ 98.8%

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

      5. rec-exp 97.2%

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

   9. Simplified 97.2%

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

       1. expm1-log1p-u 48.4%

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

       2. expm1-undefine 48.4%

          \[\leadsto e^{\log \left(e^{-x} + \color{blue}{\left(e^{\mathsf{log1p}\left(e^{x} + -2\right)} - 1\right)}\right)} \]

   11. Applied egg-rr 48.4%

       \[\leadsto e^{\log \left(e^{-x} + \color{blue}{\left(e^{\mathsf{log1p}\left(e^{x} + -2\right)} - 1\right)}\right)} \]
   12. Step-by-step derivation

       1. sub-neg 48.4%

          \[\leadsto e^{\log \left(e^{-x} + \color{blue}{\left(e^{\mathsf{log1p}\left(e^{x} + -2\right)} + \left(-1\right)\right)}\right)} \]

       2. log1p-undefine 48.4%

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

       3. rem-exp-log 97.2%

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

       4. +-commutative 97.2%

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

       5. associate-+r+ 97.2%

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

       6. metadata-eval 97.2%

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

       7. +-commutative 97.2%

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

       8. metadata-eval 97.2%

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

       9. +-commutative 97.2%

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

       10. metadata-eval 97.2%

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

       11. sub-neg 97.2%

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

       12. expm1-define 98.8%

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

   13. Simplified 98.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 0.01:\\ \;\;\;\;{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + {x}^{2} \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(e^{-x} + \left(-1 + \mathsf{expm1}\left(x\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(e^{x} - 2\right) + t\_0 \leq 0.0005:\\ \;\;\;\;{x}^{2} + {x}^{4} \cdot \mathsf{fma}\left(0.002777777777777778, {x}^{2}, 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(t\_0 + \left(-1 + \mathsf{expm1}\left(x\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (+ (- (exp x) 2.0) t_0) 0.0005)
     (+
      (pow x 2.0)
      (*
       (pow x 4.0)
       (fma 0.002777777777777778 (pow x 2.0) 0.08333333333333333)))
     (exp (log (+ t_0 (+ -1.0 (expm1 x))))))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (((exp(x) - 2.0) + t_0) <= 0.0005) {
		tmp = pow(x, 2.0) + (pow(x, 4.0) * fma(0.002777777777777778, pow(x, 2.0), 0.08333333333333333));
	} else {
		tmp = exp(log((t_0 + (-1.0 + expm1(x)))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(Float64(exp(x) - 2.0) + t_0) <= 0.0005)
		tmp = Float64((x ^ 2.0) + Float64((x ^ 4.0) * fma(0.002777777777777778, (x ^ 2.0), 0.08333333333333333)));
	else
		tmp = exp(log(Float64(t_0 + Float64(-1.0 + expm1(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], 0.0005], N[(N[Power[x, 2.0], $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * N[(0.002777777777777778 * N[Power[x, 2.0], $MachinePrecision] + 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[Log[N[(t$95$0 + N[(-1.0 + N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x))) < 5.0000000000000001e-4

   1. Initial program 55.9%

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

      1. associate-+l- 55.9%

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

      2. sub-neg 55.9%

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

      3. sub-neg 55.9%

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

      4. distribute-neg-in 55.9%

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

      5. remove-double-neg 55.9%

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

      6. +-commutative 55.9%

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

      7. metadata-eval 55.9%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + 4.96031746031746 \cdot 10^{-5} \cdot {x}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + \color{blue}{{x}^{2} \cdot 4.96031746031746 \cdot 10^{-5}}\right)\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + {x}^{2} \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right)} \]

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {x}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{{x}^{2} \cdot 1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {x}^{2}\right)\right)} \]

      2. *-rgt-identity 100.0%

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

      3. associate-*r* 100.0%

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

      4. pow-sqr 100.0%

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

      5. metadata-eval 100.0%

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

      6. +-commutative 100.0%

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

      7. fma-define 100.0%

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

   10. Simplified 100.0%

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

   if 5.0000000000000001e-4 < (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x)))

   1. Initial program 97.2%

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

      1. associate-+l- 97.3%

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

      2. sub-neg 97.3%

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

      3. sub-neg 97.3%

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

      4. distribute-neg-in 97.3%

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

      5. remove-double-neg 97.3%

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

      6. +-commutative 97.3%

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

      7. metadata-eval 97.3%

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

   3. Simplified 97.3%

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

      1. +-commutative 97.3%

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

      2. associate-+r+ 97.2%

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

      3. metadata-eval 97.2%

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

      4. sub-neg 97.2%

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

      5. add-exp-log 96.2%

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

      6. +-commutative 96.2%

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

      7. sub-neg 96.2%

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

      8. metadata-eval 96.2%

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

      9. associate-+r+ 95.7%

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

      10. +-commutative 95.7%

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

      11. +-commutative 95.7%

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

      12. cosh-undef 95.7%

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

   6. Applied egg-rr 95.7%

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

   7. Taylor expanded in x around inf 95.6%

      \[\leadsto e^{\color{blue}{\log \left(\left(e^{x} + \frac{1}{e^{x}}\right) - 2\right)}} \]
   8. Step-by-step derivation

      1. sub-neg 95.6%

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

      2. +-commutative 95.6%

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

      3. metadata-eval 95.6%

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

      4. associate-+l+ 97.8%

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

      5. rec-exp 96.2%

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

   9. Simplified 96.2%

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

       1. expm1-log1p-u 54.4%

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

       2. expm1-undefine 54.4%

          \[\leadsto e^{\log \left(e^{-x} + \color{blue}{\left(e^{\mathsf{log1p}\left(e^{x} + -2\right)} - 1\right)}\right)} \]

   11. Applied egg-rr 54.4%

       \[\leadsto e^{\log \left(e^{-x} + \color{blue}{\left(e^{\mathsf{log1p}\left(e^{x} + -2\right)} - 1\right)}\right)} \]
   12. Step-by-step derivation

       1. sub-neg 54.4%

          \[\leadsto e^{\log \left(e^{-x} + \color{blue}{\left(e^{\mathsf{log1p}\left(e^{x} + -2\right)} + \left(-1\right)\right)}\right)} \]

       2. log1p-undefine 54.4%

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

       3. rem-exp-log 96.2%

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

       4. +-commutative 96.2%

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

       5. associate-+r+ 96.2%

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

       6. metadata-eval 96.2%

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

       7. +-commutative 96.2%

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

       8. metadata-eval 96.2%

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

       9. +-commutative 96.2%

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

       10. metadata-eval 96.2%

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

       11. sub-neg 96.2%

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

       12. expm1-define 97.5%

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

   13. Simplified 97.5%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 0.0005:\\ \;\;\;\;{x}^{2} + {x}^{4} \cdot \mathsf{fma}\left(0.002777777777777778, {x}^{2}, 0.08333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(e^{-x} + \left(-1 + \mathsf{expm1}\left(x\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(e^{x} - 2\right) + t\_0 \leq 0.0005:\\ \;\;\;\;{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot 0.002777777777777778\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(t\_0 + \left(-1 + \mathsf{expm1}\left(x\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (+ (- (exp x) 2.0) t_0) 0.0005)
     (*
      (pow x 2.0)
      (+
       1.0
       (*
        (pow x 2.0)
        (+ 0.08333333333333333 (* (pow x 2.0) 0.002777777777777778)))))
     (exp (log (+ t_0 (+ -1.0 (expm1 x))))))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (((exp(x) - 2.0) + t_0) <= 0.0005) {
		tmp = pow(x, 2.0) * (1.0 + (pow(x, 2.0) * (0.08333333333333333 + (pow(x, 2.0) * 0.002777777777777778))));
	} else {
		tmp = exp(log((t_0 + (-1.0 + expm1(x)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (((Math.exp(x) - 2.0) + t_0) <= 0.0005) {
		tmp = Math.pow(x, 2.0) * (1.0 + (Math.pow(x, 2.0) * (0.08333333333333333 + (Math.pow(x, 2.0) * 0.002777777777777778))));
	} else {
		tmp = Math.exp(Math.log((t_0 + (-1.0 + Math.expm1(x)))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.exp(-x)
	tmp = 0
	if ((math.exp(x) - 2.0) + t_0) <= 0.0005:
		tmp = math.pow(x, 2.0) * (1.0 + (math.pow(x, 2.0) * (0.08333333333333333 + (math.pow(x, 2.0) * 0.002777777777777778))))
	else:
		tmp = math.exp(math.log((t_0 + (-1.0 + math.expm1(x)))))
	return tmp

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(Float64(exp(x) - 2.0) + t_0) <= 0.0005)
		tmp = Float64((x ^ 2.0) * Float64(1.0 + Float64((x ^ 2.0) * Float64(0.08333333333333333 + Float64((x ^ 2.0) * 0.002777777777777778)))));
	else
		tmp = exp(log(Float64(t_0 + Float64(-1.0 + expm1(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], 0.0005], N[(N[Power[x, 2.0], $MachinePrecision] * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.08333333333333333 + N[(N[Power[x, 2.0], $MachinePrecision] * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[Log[N[(t$95$0 + N[(-1.0 + N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x))) < 5.0000000000000001e-4

   1. Initial program 55.9%

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

      1. associate-+l- 55.9%

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

      2. sub-neg 55.9%

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

      3. sub-neg 55.9%

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

      4. distribute-neg-in 55.9%

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

      5. remove-double-neg 55.9%

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

      6. +-commutative 55.9%

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

      7. metadata-eval 55.9%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {x}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + \color{blue}{{x}^{2} \cdot 0.002777777777777778}\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot 0.002777777777777778\right)\right)} \]

   if 5.0000000000000001e-4 < (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x)))

   1. Initial program 97.2%

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

      1. associate-+l- 97.3%

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

      2. sub-neg 97.3%

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

      3. sub-neg 97.3%

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

      4. distribute-neg-in 97.3%

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

      5. remove-double-neg 97.3%

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

      6. +-commutative 97.3%

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

      7. metadata-eval 97.3%

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

   3. Simplified 97.3%

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

      1. +-commutative 97.3%

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

      2. associate-+r+ 97.2%

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

      3. metadata-eval 97.2%

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

      4. sub-neg 97.2%

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

      5. add-exp-log 96.2%

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

      6. +-commutative 96.2%

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

      7. sub-neg 96.2%

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

      8. metadata-eval 96.2%

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

      9. associate-+r+ 95.7%

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

      10. +-commutative 95.7%

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

      11. +-commutative 95.7%

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

      12. cosh-undef 95.7%

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

   6. Applied egg-rr 95.7%

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

   7. Taylor expanded in x around inf 95.6%

      \[\leadsto e^{\color{blue}{\log \left(\left(e^{x} + \frac{1}{e^{x}}\right) - 2\right)}} \]
   8. Step-by-step derivation

      1. sub-neg 95.6%

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

      2. +-commutative 95.6%

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

      3. metadata-eval 95.6%

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

      4. associate-+l+ 97.8%

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

      5. rec-exp 96.2%

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

   9. Simplified 96.2%

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

       1. expm1-log1p-u 54.4%

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

       2. expm1-undefine 54.4%

          \[\leadsto e^{\log \left(e^{-x} + \color{blue}{\left(e^{\mathsf{log1p}\left(e^{x} + -2\right)} - 1\right)}\right)} \]

   11. Applied egg-rr 54.4%

       \[\leadsto e^{\log \left(e^{-x} + \color{blue}{\left(e^{\mathsf{log1p}\left(e^{x} + -2\right)} - 1\right)}\right)} \]
   12. Step-by-step derivation

       1. sub-neg 54.4%

          \[\leadsto e^{\log \left(e^{-x} + \color{blue}{\left(e^{\mathsf{log1p}\left(e^{x} + -2\right)} + \left(-1\right)\right)}\right)} \]

       2. log1p-undefine 54.4%

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

       3. rem-exp-log 96.2%

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

       4. +-commutative 96.2%

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

       5. associate-+r+ 96.2%

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

       6. metadata-eval 96.2%

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

       7. +-commutative 96.2%

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

       8. metadata-eval 96.2%

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

       9. +-commutative 96.2%

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

       10. metadata-eval 96.2%

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

       11. sub-neg 96.2%

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

       12. expm1-define 97.5%

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

   13. Simplified 97.5%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 0.0005:\\ \;\;\;\;{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot 0.002777777777777778\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(e^{-x} + \left(-1 + \mathsf{expm1}\left(x\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(e^{x} - 2\right) + t\_0 \leq 0.0005:\\ \;\;\;\;{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot 0.002777777777777778\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(t\_0 + -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (+ (- (exp x) 2.0) t_0) 0.0005)
     (*
      (pow x 2.0)
      (+
       1.0
       (*
        (pow x 2.0)
        (+ 0.08333333333333333 (* (pow x 2.0) 0.002777777777777778)))))
     (+ (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) <= 0.0005) {
		tmp = pow(x, 2.0) * (1.0 + (pow(x, 2.0) * (0.08333333333333333 + (pow(x, 2.0) * 0.002777777777777778))));
	} 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) <= 0.0005d0) then
        tmp = (x ** 2.0d0) * (1.0d0 + ((x ** 2.0d0) * (0.08333333333333333d0 + ((x ** 2.0d0) * 0.002777777777777778d0))))
    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) <= 0.0005) {
		tmp = Math.pow(x, 2.0) * (1.0 + (Math.pow(x, 2.0) * (0.08333333333333333 + (Math.pow(x, 2.0) * 0.002777777777777778))));
	} 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) <= 0.0005:
		tmp = math.pow(x, 2.0) * (1.0 + (math.pow(x, 2.0) * (0.08333333333333333 + (math.pow(x, 2.0) * 0.002777777777777778))))
	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) <= 0.0005)
		tmp = Float64((x ^ 2.0) * Float64(1.0 + Float64((x ^ 2.0) * Float64(0.08333333333333333 + Float64((x ^ 2.0) * 0.002777777777777778)))));
	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) <= 0.0005)
		tmp = (x ^ 2.0) * (1.0 + ((x ^ 2.0) * (0.08333333333333333 + ((x ^ 2.0) * 0.002777777777777778))));
	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], 0.0005], N[(N[Power[x, 2.0], $MachinePrecision] * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.08333333333333333 + N[(N[Power[x, 2.0], $MachinePrecision] * 0.002777777777777778), $MachinePrecision]), $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) #s(literal 2 binary64)) (exp.f64 (neg.f64 x))) < 5.0000000000000001e-4

   1. Initial program 55.9%

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

      1. associate-+l- 55.9%

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

      2. sub-neg 55.9%

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

      3. sub-neg 55.9%

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

      4. distribute-neg-in 55.9%

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

      5. remove-double-neg 55.9%

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

      6. +-commutative 55.9%

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

      7. metadata-eval 55.9%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {x}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + \color{blue}{{x}^{2} \cdot 0.002777777777777778}\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot 0.002777777777777778\right)\right)} \]

   if 5.0000000000000001e-4 < (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x)))

   1. Initial program 97.2%

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

      1. associate-+l- 97.3%

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

      2. sub-neg 97.3%

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

      3. sub-neg 97.3%

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

      4. distribute-neg-in 97.3%

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

      5. remove-double-neg 97.3%

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

      6. +-commutative 97.3%

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

      7. metadata-eval 97.3%

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

   3. Simplified 97.3%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 0.0005:\\ \;\;\;\;{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot 0.002777777777777778\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(e^{-x} + -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(e^{x} - 2\right) + t\_0 \leq 10^{-9}:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(t\_0 + -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (+ (- (exp x) 2.0) t_0) 1e-9)
     (pow x 2.0)
     (+ (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) <= 1e-9) {
		tmp = pow(x, 2.0);
	} 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) <= 1d-9) then
        tmp = x ** 2.0d0
    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) <= 1e-9) {
		tmp = Math.pow(x, 2.0);
	} 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) <= 1e-9:
		tmp = math.pow(x, 2.0)
	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) <= 1e-9)
		tmp = x ^ 2.0;
	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) <= 1e-9)
		tmp = x ^ 2.0;
	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], 1e-9], N[Power[x, 2.0], $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) #s(literal 2 binary64)) (exp.f64 (neg.f64 x))) < 1.00000000000000006e-9

   1. Initial program 55.7%

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

      1. associate-+l- 55.7%

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

      2. sub-neg 55.7%

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

      3. sub-neg 55.7%

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

      4. distribute-neg-in 55.7%

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

      5. remove-double-neg 55.7%

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

      6. +-commutative 55.7%

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

      7. metadata-eval 55.7%

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

   3. Simplified 55.7%

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

   5. Taylor expanded in x around 0 99.8%

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

   if 1.00000000000000006e-9 < (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x)))

   1. Initial program 96.2%

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

      1. associate-+l- 96.3%

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

      2. sub-neg 96.3%

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

      3. sub-neg 96.3%

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

      4. distribute-neg-in 96.3%

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

      5. remove-double-neg 96.3%

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

      6. +-commutative 96.3%

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

      7. metadata-eval 96.3%

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

   3. Simplified 96.3%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 10^{-9}:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(e^{-x} + -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(e^{x} - 2\right) + t\_0 \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(t\_0 + -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (+ (- (exp x) 2.0) t_0) 1e-9)
     (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) <= 1e-9) {
		tmp = 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) <= 1e-9)
		tmp = 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], 1e-9], N[(x * x + 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) #s(literal 2 binary64)) (exp.f64 (neg.f64 x))) < 1.00000000000000006e-9

   1. Initial program 55.7%

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

      1. associate-+l- 55.7%

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

      2. sub-neg 55.7%

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

      3. sub-neg 55.7%

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

      4. distribute-neg-in 55.7%

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

      5. remove-double-neg 55.7%

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

      6. +-commutative 55.7%

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

      7. metadata-eval 55.7%

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

   3. Simplified 55.7%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + 4.96031746031746 \cdot 10^{-5} \cdot {x}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + \color{blue}{{x}^{2} \cdot 4.96031746031746 \cdot 10^{-5}}\right)\right)\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot \left(0.002777777777777778 + {x}^{2} \cdot 4.96031746031746 \cdot 10^{-5}\right)\right)\right)} \]

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {x}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto {x}^{2} \cdot \left(1 + \color{blue}{{x}^{2} \cdot 0.08333333333333333}\right) \]

      2. distribute-lft-in 100.0%

         \[\leadsto \color{blue}{{x}^{2} \cdot 1 + {x}^{2} \cdot \left({x}^{2} \cdot 0.08333333333333333\right)} \]

      3. *-rgt-identity 100.0%

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

      4. associate-*r* 100.0%

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

      5. pow-sqr 100.0%

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

      6. metadata-eval 100.0%

         \[\leadsto {x}^{2} + {x}^{\color{blue}{4}} \cdot 0.08333333333333333 \]

   10. Simplified 100.0%

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

       1. unpow2 100.0%

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

       2. fma-define 100.0%

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

   12. Applied egg-rr 100.0%

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

   if 1.00000000000000006e-9 < (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x)))

   1. Initial program 96.2%

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

      1. associate-+l- 96.3%

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

      2. sub-neg 96.3%

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

      3. sub-neg 96.3%

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

      4. distribute-neg-in 96.3%

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

      5. remove-double-neg 96.3%

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

      6. +-commutative 96.3%

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

      7. metadata-eval 96.3%

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

   3. Simplified 96.3%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(e^{-x} + -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.000116:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.000116) (pow x 2.0) (- (* 2.0 (cosh x)) 2.0)))

C

double code(double x) {
	double tmp;
	if (x <= 0.000116) {
		tmp = pow(x, 2.0);
	} else {
		tmp = (2.0 * cosh(x)) - 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.000116d0) then
        tmp = x ** 2.0d0
    else
        tmp = (2.0d0 * cosh(x)) - 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.000116) {
		tmp = Math.pow(x, 2.0);
	} else {
		tmp = (2.0 * Math.cosh(x)) - 2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.000116:
		tmp = math.pow(x, 2.0)
	else:
		tmp = (2.0 * math.cosh(x)) - 2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.000116)
		tmp = x ^ 2.0;
	else
		tmp = Float64(Float64(2.0 * cosh(x)) - 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.000116)
		tmp = x ^ 2.0;
	else
		tmp = (2.0 * cosh(x)) - 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.000116], N[Power[x, 2.0], $MachinePrecision], N[(N[(2.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.16e-4

   1. Initial program 56.3%

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

      1. associate-+l- 56.3%

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

      2. sub-neg 56.3%

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

      3. sub-neg 56.3%

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

      4. distribute-neg-in 56.3%

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

      5. remove-double-neg 56.3%

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

      6. +-commutative 56.3%

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

      7. metadata-eval 56.3%

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

   3. Simplified 56.3%

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

   5. Taylor expanded in x around 0 98.8%

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

   if 1.16e-4 < x

   1. Initial program 93.9%

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

      1. associate-+l- 94.0%

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

      2. sub-neg 94.0%

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

      3. sub-neg 94.0%

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

      4. distribute-neg-in 94.0%

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

      5. remove-double-neg 94.0%

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

      6. +-commutative 94.0%

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

      7. metadata-eval 94.0%

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

   3. Simplified 94.0%

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

      1. +-commutative 94.0%

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

      2. associate-+r+ 93.9%

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

      3. metadata-eval 93.9%

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

      4. sub-neg 93.9%

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

      5. +-commutative 93.9%

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

      6. associate-+r- 93.2%

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

      7. +-commutative 93.2%

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

      8. cosh-undef 93.3%

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

   6. Applied egg-rr 93.3%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000116:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.5% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (pow x 2.0))

C

double code(double x) {
	return pow(x, 2.0);
}

Fortran

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

Java

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

Python

def code(x):
	return math.pow(x, 2.0)

Julia

function code(x)
	return x ^ 2.0
end

MATLAB

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

Wolfram

code[x_] := N[Power[x, 2.0], $MachinePrecision]

Derivation

1. Initial program 57.0%

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

   1. associate-+l- 57.0%

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

   2. sub-neg 57.0%

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

   3. sub-neg 57.0%

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

   4. distribute-neg-in 57.0%

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

   5. remove-double-neg 57.0%

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

   6. +-commutative 57.0%

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

   7. metadata-eval 57.0%

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

3. Simplified 57.0%

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

5. Taylor expanded in x around 0 97.3%

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

6. Final simplification 97.3%

   \[\leadsto {x}^{2} \]
7. Add Preprocessing

Alternative 9: 6.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (expm1 x))

C

double code(double x) {
	return expm1(x);
}

Java

public static double code(double x) {
	return Math.expm1(x);
}

Python

def code(x):
	return math.expm1(x)

Julia

function code(x)
	return expm1(x)
end

Wolfram

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

Derivation

1. Initial program 57.0%

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

   1. associate-+l- 57.0%

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

   2. sub-neg 57.0%

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

   3. sub-neg 57.0%

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

   4. distribute-neg-in 57.0%

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

   5. remove-double-neg 57.0%

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

   6. +-commutative 57.0%

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

   7. metadata-eval 57.0%

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

3. Simplified 57.0%

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

5. Taylor expanded in x around 0 54.6%

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

6. Taylor expanded in x around inf 54.6%

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

   1. expm1-define 6.8%

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

8. Simplified 6.8%

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

9. Final simplification 6.8%

   \[\leadsto \mathsf{expm1}\left(x\right) \]
10. Add Preprocessing

Alternative 10: 6.0% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (+
   1.0
   (* x (+ 0.5 (* x (+ 0.16666666666666666 (* x 0.041666666666666664))))))))

C

double code(double x) {
	return x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + (x * (0.5d0 + (x * (0.16666666666666666d0 + (x * 0.041666666666666664d0))))))
end function

Java

public static double code(double x) {
	return x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))));
}

Python

def code(x):
	return x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(0.16666666666666666 + Float64(x * 0.041666666666666664)))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664))))));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * N[(0.16666666666666666 + N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.0%

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

   1. associate-+l- 57.0%

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

   2. sub-neg 57.0%

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

   3. sub-neg 57.0%

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

   4. distribute-neg-in 57.0%

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

   5. remove-double-neg 57.0%

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

   6. +-commutative 57.0%

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

   7. metadata-eval 57.0%

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

3. Simplified 57.0%

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

5. Taylor expanded in x around 0 54.6%

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

6. Taylor expanded in x around 0 6.2%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + 0.041666666666666664 \cdot x\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative 6.2%

      \[\leadsto x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + \color{blue}{x \cdot 0.041666666666666664}\right)\right)\right) \]

8. Simplified 6.2%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right)} \]

9. Final simplification 6.2%

   \[\leadsto x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\right) \]
10. Add Preprocessing

Alternative 11: 5.9% accurate, 18.7× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666))))))

C

double code(double x) {
	return x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + (x * (0.5d0 + (x * 0.16666666666666666d0))))
end function

Java

public static double code(double x) {
	return x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))));
}

Python

def code(x):
	return x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666)))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.0%

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

   1. associate-+l- 57.0%

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

   2. sub-neg 57.0%

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

   3. sub-neg 57.0%

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

   4. distribute-neg-in 57.0%

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

   5. remove-double-neg 57.0%

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

   6. +-commutative 57.0%

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

   7. metadata-eval 57.0%

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

3. Simplified 57.0%

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

5. Taylor expanded in x around 0 54.6%

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

6. Taylor expanded in x around 0 6.1%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(0.5 + 0.16666666666666666 \cdot x\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative 6.1%

      \[\leadsto x \cdot \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.16666666666666666}\right)\right) \]

8. Simplified 6.1%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)} \]

9. Final simplification 6.1%

   \[\leadsto x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right) \]
10. Add Preprocessing

Alternative 12: 5.9% accurate, 29.4× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + x \cdot 0.5\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.0%

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

   1. associate-+l- 57.0%

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

   2. sub-neg 57.0%

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

   3. sub-neg 57.0%

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

   4. distribute-neg-in 57.0%

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

   5. remove-double-neg 57.0%

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

   6. +-commutative 57.0%

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

   7. metadata-eval 57.0%

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

3. Simplified 57.0%

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

5. Taylor expanded in x around 0 54.6%

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

6. Taylor expanded in x around 0 6.2%

   \[\leadsto \color{blue}{x \cdot \left(1 + 0.5 \cdot x\right)} \]
7. Step-by-step derivation

   1. *-commutative 6.2%

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

8. Simplified 6.2%

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

9. Final simplification 6.2%

   \[\leadsto x \cdot \left(1 + x \cdot 0.5\right) \]
10. Add Preprocessing

Alternative 13: 5.9% 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 57.0%

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

   1. associate-+l- 57.0%

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

   2. sub-neg 57.0%

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

   3. sub-neg 57.0%

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

   4. distribute-neg-in 57.0%

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

   5. remove-double-neg 57.0%

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

   6. +-commutative 57.0%

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

   7. metadata-eval 57.0%

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

3. Simplified 57.0%

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

5. Taylor expanded in x around 0 54.6%

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

6. Taylor expanded in x around 0 6.1%

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

7. Final simplification 6.1%

   \[\leadsto x \]
8. Add Preprocessing

Developer target: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sinh \left(\frac{x}{2}\right)\\ 4 \cdot \left(t\_0 \cdot t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sinh (/ x 2.0)))) (* 4.0 (* t_0 t_0))))

C

double code(double x) {
	double t_0 = sinh((x / 2.0));
	return 4.0 * (t_0 * t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sinh((x / 2.0d0))
    code = 4.0d0 * (t_0 * t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.sinh((x / 2.0));
	return 4.0 * (t_0 * t_0);
}

Python

def code(x):
	t_0 = math.sinh((x / 2.0))
	return 4.0 * (t_0 * t_0)

Julia

function code(x)
	t_0 = sinh(Float64(x / 2.0))
	return Float64(4.0 * Float64(t_0 * t_0))
end

MATLAB

function tmp = code(x)
	t_0 = sinh((x / 2.0));
	tmp = 4.0 * (t_0 * t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sinh[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024066 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64
  :pre (<= (fabs x) 710.0)

  :alt
  (* 4.0 (* (sinh (/ x 2.0)) (sinh (/ x 2.0))))

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