exp2 (problem 3.3.7)

Percentage Accurate: 10.4% → 99.9%

Time: 38.4s

Alternatives: 10

Speedup: 1.9×

• could not determine a ground truth

  1. x = -8.420938327346457e-292

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: 10.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(e^{x\_m} - 2\right) + e^{-x\_m}\\ \mathbf{if}\;t\_0 \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, {x\_m}^{6} \cdot \left(0.002777777777777778 + \frac{0.08333333333333333}{{x\_m}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ (- (exp x_m) 2.0) (exp (- x_m)))))
   (if (<= t_0 0.0001)
     (fma
      x_m
      x_m
      (*
       (pow x_m 6.0)
       (+ 0.002777777777777778 (/ 0.08333333333333333 (pow x_m 2.0)))))
     t_0)))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = (exp(x_m) - 2.0) + exp(-x_m);
	double tmp;
	if (t_0 <= 0.0001) {
		tmp = fma(x_m, x_m, (pow(x_m, 6.0) * (0.002777777777777778 + (0.08333333333333333 / pow(x_m, 2.0)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(exp(x_m) - 2.0) + exp(Float64(-x_m)))
	tmp = 0.0
	if (t_0 <= 0.0001)
		tmp = fma(x_m, x_m, Float64((x_m ^ 6.0) * Float64(0.002777777777777778 + Float64(0.08333333333333333 / (x_m ^ 2.0)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[x$95$m], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x$95$m)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0001], N[(x$95$m * x$95$m + N[(N[Power[x$95$m, 6.0], $MachinePrecision] * N[(0.002777777777777778 + N[(0.08333333333333333 / N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x))) < 1.00000000000000005e-4

   1. Initial program 6.9%

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

      1. associate-+l- 6.9%

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

      2. sub-neg 6.9%

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

      3. sub-neg 6.9%

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

      4. distribute-neg-in 6.9%

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

      5. remove-double-neg 6.9%

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

      6. +-commutative 6.9%

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

      7. metadata-eval 6.9%

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

   3. Simplified 6.9%

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

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

      1. distribute-rgt-in 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. unpow2 100.0%

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

      4. fma-define 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-*r* 100.0%

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

      9. pow-prod-up 100.0%

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

      10. metadata-eval 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around inf 100.0%

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

       1. associate-*r/ 100.0%

          \[\leadsto \mathsf{fma}\left(x, x, {x}^{6} \cdot \left(0.002777777777777778 + \color{blue}{\frac{0.08333333333333333 \cdot 1}{{x}^{2}}}\right)\right) \]

       2. metadata-eval 100.0%

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

   12. Simplified 100.0%

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

   if 1.00000000000000005e-4 < (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x)))

   1. Initial program 99.1%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(x, x, {x}^{6} \cdot \left(0.002777777777777778 + \frac{0.08333333333333333}{{x}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x} - 2\right) + e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{-x\_m}\\ \mathbf{if}\;\left(e^{x\_m} - 2\right) + t\_0 \leq 5 \cdot 10^{-9}:\\ \;\;\;\;{x\_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;e^{x\_m} + \left(t\_0 + -2\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (exp (- x_m))))
   (if (<= (+ (- (exp x_m) 2.0) t_0) 5e-9)
     (pow x_m 2.0)
     (+ (exp x_m) (+ t_0 -2.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = exp(-x_m);
	double tmp;
	if (((exp(x_m) - 2.0) + t_0) <= 5e-9) {
		tmp = pow(x_m, 2.0);
	} else {
		tmp = exp(x_m) + (t_0 + -2.0);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x_m)
    if (((exp(x_m) - 2.0d0) + t_0) <= 5d-9) then
        tmp = x_m ** 2.0d0
    else
        tmp = exp(x_m) + (t_0 + (-2.0d0))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = Math.exp(-x_m);
	double tmp;
	if (((Math.exp(x_m) - 2.0) + t_0) <= 5e-9) {
		tmp = Math.pow(x_m, 2.0);
	} else {
		tmp = Math.exp(x_m) + (t_0 + -2.0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = math.exp(-x_m)
	tmp = 0
	if ((math.exp(x_m) - 2.0) + t_0) <= 5e-9:
		tmp = math.pow(x_m, 2.0)
	else:
		tmp = math.exp(x_m) + (t_0 + -2.0)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = exp(Float64(-x_m))
	tmp = 0.0
	if (Float64(Float64(exp(x_m) - 2.0) + t_0) <= 5e-9)
		tmp = x_m ^ 2.0;
	else
		tmp = Float64(exp(x_m) + Float64(t_0 + -2.0));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = exp(-x_m);
	tmp = 0.0;
	if (((exp(x_m) - 2.0) + t_0) <= 5e-9)
		tmp = x_m ^ 2.0;
	else
		tmp = exp(x_m) + (t_0 + -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Exp[(-x$95$m)], $MachinePrecision]}, If[LessEqual[N[(N[(N[Exp[x$95$m], $MachinePrecision] - 2.0), $MachinePrecision] + t$95$0), $MachinePrecision], 5e-9], N[Power[x$95$m, 2.0], $MachinePrecision], N[(N[Exp[x$95$m], $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-9

   1. Initial program 6.1%

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

      1. associate-+l- 6.0%

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

      2. sub-neg 6.0%

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

      3. sub-neg 6.0%

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

      4. distribute-neg-in 6.0%

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

      5. remove-double-neg 6.0%

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

      6. +-commutative 6.0%

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

      7. metadata-eval 6.0%

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

   3. Simplified 6.0%

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

   5. Taylor expanded in x around 0 99.3%

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

   if 5.0000000000000001e-9 < (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x)))

   1. Initial program 93.1%

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

      1. associate-+l- 94.1%

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

      2. sub-neg 94.1%

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

      3. sub-neg 94.1%

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

      4. distribute-neg-in 94.1%

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

      5. remove-double-neg 94.1%

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

      6. +-commutative 94.1%

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

      7. metadata-eval 94.1%

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

   3. Simplified 94.1%

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

4. Final simplification 99.0%

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

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(e^{x\_m} - 2\right) + e^{-x\_m}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, x\_m, 0.08333333333333333 \cdot {x\_m}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ (- (exp x_m) 2.0) (exp (- x_m)))))
   (if (<= t_0 2e-5) (fma x_m x_m (* 0.08333333333333333 (pow x_m 4.0))) t_0)))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = (exp(x_m) - 2.0) + exp(-x_m);
	double tmp;
	if (t_0 <= 2e-5) {
		tmp = fma(x_m, x_m, (0.08333333333333333 * pow(x_m, 4.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x))) < 2.00000000000000016e-5

   1. Initial program 6.6%

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

      1. associate-+l- 6.6%

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

      2. sub-neg 6.6%

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

      3. sub-neg 6.6%

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

      4. distribute-neg-in 6.6%

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

      5. remove-double-neg 6.6%

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

      6. +-commutative 6.6%

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

      7. metadata-eval 6.6%

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

   3. Simplified 6.6%

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

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

      1. distribute-rgt-in 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. unpow2 100.0%

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

      4. fma-define 100.0%

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

      5. +-commutative 100.0%

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

      6. fma-undefine 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-*r* 100.0%

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

      9. pow-prod-up 100.0%

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

      10. metadata-eval 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around 0 99.9%

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

   if 2.00000000000000016e-5 < (+.f64 (-.f64 (exp.f64 x) #s(literal 2 binary64)) (exp.f64 (neg.f64 x)))

   1. Initial program 97.4%

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

4. Final simplification 99.8%

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

Alternative 4: 98.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0002:\\ \;\;\;\;{x\_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x\_m - 2\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0002) (pow x_m 2.0) (- (* 2.0 (cosh x_m)) 2.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0002) {
		tmp = pow(x_m, 2.0);
	} else {
		tmp = (2.0 * cosh(x_m)) - 2.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.0002d0) then
        tmp = x_m ** 2.0d0
    else
        tmp = (2.0d0 * cosh(x_m)) - 2.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.0002) {
		tmp = Math.pow(x_m, 2.0);
	} else {
		tmp = (2.0 * Math.cosh(x_m)) - 2.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.0002:
		tmp = math.pow(x_m, 2.0)
	else:
		tmp = (2.0 * math.cosh(x_m)) - 2.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0002)
		tmp = x_m ^ 2.0;
	else
		tmp = Float64(Float64(2.0 * cosh(x_m)) - 2.0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.0002)
		tmp = x_m ^ 2.0;
	else
		tmp = (2.0 * cosh(x_m)) - 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0002], N[Power[x$95$m, 2.0], $MachinePrecision], N[(N[(2.0 * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e-4

   1. Initial program 8.9%

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

      1. associate-+l- 8.9%

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

      2. sub-neg 8.9%

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

      3. sub-neg 8.9%

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

      4. distribute-neg-in 8.9%

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

      5. remove-double-neg 8.9%

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

      6. +-commutative 8.9%

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

      7. metadata-eval 8.9%

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

   3. Simplified 8.9%

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

   5. Taylor expanded in x around 0 96.8%

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

   if 2.0000000000000001e-4 < x

   1. Initial program 91.2%

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

      1. associate-+l- 92.5%

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

      2. sub-neg 92.5%

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

      3. sub-neg 92.5%

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

      4. distribute-neg-in 92.5%

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

      5. remove-double-neg 92.5%

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

      6. +-commutative 92.5%

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

      7. metadata-eval 92.5%

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

   3. Simplified 92.5%

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

      1. +-commutative 92.5%

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

      2. associate-+r+ 91.2%

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

      3. metadata-eval 91.2%

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

      4. sub-neg 91.2%

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

      5. +-commutative 91.2%

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

      6. associate-+r- 89.9%

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

      7. +-commutative 89.9%

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

      8. cosh-undef 90.1%

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

   6. Applied egg-rr 90.1%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0002:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x - 2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.65:\\ \;\;\;\;{x\_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\_m\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 1.65) (pow x_m 2.0) (expm1 x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.65) {
		tmp = pow(x_m, 2.0);
	} else {
		tmp = expm1(x_m);
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.65) {
		tmp = Math.pow(x_m, 2.0);
	} else {
		tmp = Math.expm1(x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.65:
		tmp = math.pow(x_m, 2.0)
	else:
		tmp = math.expm1(x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.65)
		tmp = x_m ^ 2.0;
	else
		tmp = expm1(x_m);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.65], N[Power[x$95$m, 2.0], $MachinePrecision], N[(Exp[x$95$m] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6499999999999999

   1. Initial program 9.4%

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

      1. associate-+l- 9.5%

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

      2. sub-neg 9.5%

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

      3. sub-neg 9.5%

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

      4. distribute-neg-in 9.5%

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

      5. remove-double-neg 9.5%

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

      6. +-commutative 9.5%

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

      7. metadata-eval 9.5%

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

   3. Simplified 9.5%

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

   5. Taylor expanded in x around 0 96.4%

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

   if 1.6499999999999999 < x

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. remove-double-neg 100.0%

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

      6. +-commutative 100.0%

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

      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. Add Preprocessing

   5. Taylor expanded in x around 0 71.9%

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

   6. Taylor expanded in x around inf 71.9%

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

      1. expm1-define 71.9%

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

   8. Simplified 71.9%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 8.7% accurate, 2.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (expm1 x_m))

C

x_m = fabs(x);
double code(double x_m) {
	return expm1(x_m);
}

Java

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

Python

x_m = math.fabs(x)
def code(x_m):
	return math.expm1(x_m)

Julia

x_m = abs(x)
function code(x_m)
	return expm1(x_m)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(Exp[x$95$m] - 1), $MachinePrecision]

Derivation

1. Initial program 10.5%

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

   1. associate-+l- 10.5%

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

   2. sub-neg 10.5%

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

   3. sub-neg 10.5%

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

   4. distribute-neg-in 10.5%

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

   5. remove-double-neg 10.5%

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

   6. +-commutative 10.5%

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

   7. metadata-eval 10.5%

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

3. Simplified 10.5%

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

5. Taylor expanded in x around 0 5.5%

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

6. Taylor expanded in x around inf 5.5%

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

   1. expm1-define 5.5%

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

8. Simplified 5.5%

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

9. Final simplification 5.5%

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

Alternative 7: 7.5% accurate, 13.7× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  x_m
  (+
   1.0
   (*
    x_m
    (+ 0.5 (* x_m (+ 0.16666666666666666 (* x_m 0.041666666666666664))))))))

C

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

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return x_m * (1.0 + (x_m * (0.5 + (x_m * (0.16666666666666666 + (x_m * 0.041666666666666664))))));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return x_m * (1.0 + (x_m * (0.5 + (x_m * (0.16666666666666666 + (x_m * 0.041666666666666664))))))

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * N[(1.0 + N[(x$95$m * N[(0.5 + N[(x$95$m * N[(0.16666666666666666 + N[(x$95$m * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 10.5%

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

   1. associate-+l- 10.5%

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

   2. sub-neg 10.5%

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

   3. sub-neg 10.5%

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

   4. distribute-neg-in 10.5%

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

   5. remove-double-neg 10.5%

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

   6. +-commutative 10.5%

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

   7. metadata-eval 10.5%

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

3. Simplified 10.5%

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

5. Taylor expanded in x around 0 5.5%

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

6. Taylor expanded in x around 0 4.9%

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

   1. *-commutative 4.9%

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

8. Simplified 4.9%

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

9. Final simplification 4.9%

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

Alternative 8: 7.5% accurate, 18.7× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (* x_m (+ 1.0 (* x_m (+ 0.5 (* x_m 0.16666666666666666))))))

C

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

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return x_m * (1.0 + (x_m * (0.5 + (x_m * 0.16666666666666666))));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return x_m * (1.0 + (x_m * (0.5 + (x_m * 0.16666666666666666))))

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * N[(1.0 + N[(x$95$m * N[(0.5 + N[(x$95$m * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 10.5%

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

   1. associate-+l- 10.5%

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

   2. sub-neg 10.5%

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

   3. sub-neg 10.5%

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

   4. distribute-neg-in 10.5%

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

   5. remove-double-neg 10.5%

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

   6. +-commutative 10.5%

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

   7. metadata-eval 10.5%

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

3. Simplified 10.5%

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

5. Taylor expanded in x around 0 5.5%

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

6. Taylor expanded in x around 0 4.8%

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

   1. *-commutative 4.8%

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

8. Simplified 4.8%

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

9. Final simplification 4.8%

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

Alternative 9: 7.5% accurate, 29.4× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* x_m (+ 1.0 (* x_m 0.5))))

C

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

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return x_m * (1.0 + (x_m * 0.5));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return x_m * (1.0 + (x_m * 0.5))

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * N[(1.0 + N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 10.5%

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

   1. associate-+l- 10.5%

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

   2. sub-neg 10.5%

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

   3. sub-neg 10.5%

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

   4. distribute-neg-in 10.5%

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

   5. remove-double-neg 10.5%

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

   6. +-commutative 10.5%

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

   7. metadata-eval 10.5%

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

3. Simplified 10.5%

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

5. Taylor expanded in x around 0 5.5%

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

6. Taylor expanded in x around 0 4.9%

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

   1. *-commutative 4.9%

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

8. Simplified 4.9%

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

9. Final simplification 4.9%

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

Alternative 10: 7.5% accurate, 206.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x\_m \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 x_m)

C

x_m = fabs(x);
double code(double x_m) {
	return x_m;
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return x_m;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return x_m

Julia

x_m = abs(x)
function code(x_m)
	return x_m
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := x$95$m

Derivation

1. Initial program 10.5%

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

   1. associate-+l- 10.5%

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

   2. sub-neg 10.5%

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

   3. sub-neg 10.5%

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

   4. distribute-neg-in 10.5%

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

   5. remove-double-neg 10.5%

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

   6. +-commutative 10.5%

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

   7. metadata-eval 10.5%

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

3. Simplified 10.5%

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

5. Taylor expanded in x around 0 5.5%

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

6. Taylor expanded in x around 0 4.8%

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

7. Final simplification 4.8%

   \[\leadsto x \]
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 2024083 
(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))))