exp2 (problem 3.3.7)

Percentage Accurate: 77.1% → 99.9%

Time: 7.2s

Alternatives: 8

Speedup: 2.0×

• 49.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (- (exp x) 2.0) (exp (- x))) 4e-7)
   (fma
    (* x x)
    (* (* x x) 0.08333333333333333)
    (fma x x (* 0.002777777777777778 (pow x 6.0))))
   (+ (* 2.0 (cosh x)) -2.0)))

C

double code(double x) {
	double tmp;
	if (((exp(x) - 2.0) + exp(-x)) <= 4e-7) {
		tmp = fma((x * x), ((x * x) * 0.08333333333333333), fma(x, x, (0.002777777777777778 * pow(x, 6.0))));
	} else {
		tmp = (2.0 * cosh(x)) + -2.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(exp(x) - 2.0) + exp(Float64(-x))) <= 4e-7)
		tmp = fma(Float64(x * x), Float64(Float64(x * x) * 0.08333333333333333), fma(x, x, Float64(0.002777777777777778 * (x ^ 6.0))));
	else
		tmp = Float64(Float64(2.0 * cosh(x)) + -2.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 4e-7], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.08333333333333333), $MachinePrecision] + N[(x * x + N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 3.9999999999999998e-7

   1. Initial program 64.0%

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

      1. associate-+l- 64.0%

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

      2. sub-neg 64.0%

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

      3. sub-neg 64.0%

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

      4. +-commutative 64.0%

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

      5. distribute-neg-in 64.0%

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

      6. remove-double-neg 64.0%

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

      7. metadata-eval 64.0%

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

   3. Simplified 64.0%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. flip-+ 19.5%

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

      3. pow-prod-up 19.4%

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

      4. metadata-eval 19.4%

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

      5. *-commutative 19.4%

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

      6. *-commutative 19.4%

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

      7. swap-sqr 19.4%

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

      8. pow-prod-up 19.4%

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

      9. metadata-eval 19.4%

         \[\leadsto 0.002777777777777778 \cdot {x}^{6} + \frac{{x}^{4} - {x}^{\color{blue}{8}} \cdot \left(0.08333333333333333 \cdot 0.08333333333333333\right)}{{x}^{2} - 0.08333333333333333 \cdot {x}^{4}} \]

      10. metadata-eval 19.4%

          \[\leadsto 0.002777777777777778 \cdot {x}^{6} + \frac{{x}^{4} - {x}^{8} \cdot \color{blue}{0.006944444444444444}}{{x}^{2} - 0.08333333333333333 \cdot {x}^{4}} \]

      11. unpow2 19.4%

          \[\leadsto 0.002777777777777778 \cdot {x}^{6} + \frac{{x}^{4} - {x}^{8} \cdot 0.006944444444444444}{\color{blue}{x \cdot x} - 0.08333333333333333 \cdot {x}^{4}} \]

   6. Applied egg-rr 19.4%

      \[\leadsto 0.002777777777777778 \cdot {x}^{6} + \color{blue}{\frac{{x}^{4} - {x}^{8} \cdot 0.006944444444444444}{x \cdot x - 0.08333333333333333 \cdot {x}^{4}}} \]
   7. Step-by-step derivation

      1. div-sub 19.4%

         \[\leadsto 0.002777777777777778 \cdot {x}^{6} + \color{blue}{\left(\frac{{x}^{4}}{x \cdot x - 0.08333333333333333 \cdot {x}^{4}} - \frac{{x}^{8} \cdot 0.006944444444444444}{x \cdot x - 0.08333333333333333 \cdot {x}^{4}}\right)} \]

      2. *-commutative 19.4%

         \[\leadsto 0.002777777777777778 \cdot {x}^{6} + \left(\frac{{x}^{4}}{x \cdot x - 0.08333333333333333 \cdot {x}^{4}} - \frac{\color{blue}{0.006944444444444444 \cdot {x}^{8}}}{x \cdot x - 0.08333333333333333 \cdot {x}^{4}}\right) \]

      3. metadata-eval 19.4%

         \[\leadsto 0.002777777777777778 \cdot {x}^{6} + \left(\frac{{x}^{4}}{x \cdot x - 0.08333333333333333 \cdot {x}^{4}} - \frac{0.006944444444444444 \cdot {x}^{\color{blue}{\left(2 \cdot 4\right)}}}{x \cdot x - 0.08333333333333333 \cdot {x}^{4}}\right) \]

      4. pow-sqr 19.4%

         \[\leadsto 0.002777777777777778 \cdot {x}^{6} + \left(\frac{{x}^{4}}{x \cdot x - 0.08333333333333333 \cdot {x}^{4}} - \frac{0.006944444444444444 \cdot \color{blue}{\left({x}^{4} \cdot {x}^{4}\right)}}{x \cdot x - 0.08333333333333333 \cdot {x}^{4}}\right) \]

      5. metadata-eval 19.4%

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

      6. swap-sqr 19.4%

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

      7. div-sub 19.4%

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

      8. sqr-pow 19.5%

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

      9. metadata-eval 19.5%

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

      10. pow2 19.5%

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

      11. metadata-eval 19.5%

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

      12. pow2 19.5%

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

      13. flip-+ 100.0%

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

      14. +-commutative 100.0%

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

      15. *-commutative 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+l+ 100.0%

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

      3. sqr-pow 100.0%

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

      4. metadata-eval 100.0%

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

      5. pow2 100.0%

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

      6. metadata-eval 100.0%

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

      7. pow2 100.0%

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

      8. associate-*l* 100.0%

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

      9. fma-def 100.0%

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

      10. fma-def 100.0%

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

   10. Applied egg-rr 100.0%

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

   if 3.9999999999999998e-7 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+r- 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+r- 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-+l- 100.0%

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

   3. Simplified 100.0%

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

      1. associate--r- 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-+r+ 100.0%

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

      6. cosh-undef 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (- (exp x) 2.0) (exp (- x))) 4e-7)
   (+
    (* 0.002777777777777778 (pow x 6.0))
    (+ (* x x) (* 0.08333333333333333 (pow x 4.0))))
   (+ (* 2.0 (cosh x)) -2.0)))

C

double code(double x) {
	double tmp;
	if (((exp(x) - 2.0) + exp(-x)) <= 4e-7) {
		tmp = (0.002777777777777778 * pow(x, 6.0)) + ((x * x) + (0.08333333333333333 * pow(x, 4.0)));
	} 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 (((exp(x) - 2.0d0) + exp(-x)) <= 4d-7) then
        tmp = (0.002777777777777778d0 * (x ** 6.0d0)) + ((x * x) + (0.08333333333333333d0 * (x ** 4.0d0)))
    else
        tmp = (2.0d0 * cosh(x)) + (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((Math.exp(x) - 2.0) + Math.exp(-x)) <= 4e-7) {
		tmp = (0.002777777777777778 * Math.pow(x, 6.0)) + ((x * x) + (0.08333333333333333 * Math.pow(x, 4.0)));
	} else {
		tmp = (2.0 * Math.cosh(x)) + -2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((math.exp(x) - 2.0) + math.exp(-x)) <= 4e-7:
		tmp = (0.002777777777777778 * math.pow(x, 6.0)) + ((x * x) + (0.08333333333333333 * math.pow(x, 4.0)))
	else:
		tmp = (2.0 * math.cosh(x)) + -2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(exp(x) - 2.0) + exp(Float64(-x))) <= 4e-7)
		tmp = Float64(Float64(0.002777777777777778 * (x ^ 6.0)) + Float64(Float64(x * x) + Float64(0.08333333333333333 * (x ^ 4.0))));
	else
		tmp = Float64(Float64(2.0 * cosh(x)) + -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((exp(x) - 2.0) + exp(-x)) <= 4e-7)
		tmp = (0.002777777777777778 * (x ^ 6.0)) + ((x * x) + (0.08333333333333333 * (x ^ 4.0)));
	else
		tmp = (2.0 * cosh(x)) + -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 4e-7], N[(N[(0.002777777777777778 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 3.9999999999999998e-7

   1. Initial program 64.0%

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

      1. associate-+l- 64.0%

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

      2. sub-neg 64.0%

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

      3. sub-neg 64.0%

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

      4. +-commutative 64.0%

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

      5. distribute-neg-in 64.0%

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

      6. remove-double-neg 64.0%

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

      7. metadata-eval 64.0%

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

   3. Simplified 64.0%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. flip-+ 19.5%

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

      3. pow-prod-up 19.4%

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

      4. metadata-eval 19.4%

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

      5. *-commutative 19.4%

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

      6. *-commutative 19.4%

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

      7. swap-sqr 19.4%

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

      8. pow-prod-up 19.4%

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

      9. metadata-eval 19.4%

         \[\leadsto 0.002777777777777778 \cdot {x}^{6} + \frac{{x}^{4} - {x}^{\color{blue}{8}} \cdot \left(0.08333333333333333 \cdot 0.08333333333333333\right)}{{x}^{2} - 0.08333333333333333 \cdot {x}^{4}} \]

      10. metadata-eval 19.4%

          \[\leadsto 0.002777777777777778 \cdot {x}^{6} + \frac{{x}^{4} - {x}^{8} \cdot \color{blue}{0.006944444444444444}}{{x}^{2} - 0.08333333333333333 \cdot {x}^{4}} \]

      11. unpow2 19.4%

          \[\leadsto 0.002777777777777778 \cdot {x}^{6} + \frac{{x}^{4} - {x}^{8} \cdot 0.006944444444444444}{\color{blue}{x \cdot x} - 0.08333333333333333 \cdot {x}^{4}} \]

   6. Applied egg-rr 19.4%

      \[\leadsto 0.002777777777777778 \cdot {x}^{6} + \color{blue}{\frac{{x}^{4} - {x}^{8} \cdot 0.006944444444444444}{x \cdot x - 0.08333333333333333 \cdot {x}^{4}}} \]
   7. Step-by-step derivation

      1. div-sub 19.4%

         \[\leadsto 0.002777777777777778 \cdot {x}^{6} + \color{blue}{\left(\frac{{x}^{4}}{x \cdot x - 0.08333333333333333 \cdot {x}^{4}} - \frac{{x}^{8} \cdot 0.006944444444444444}{x \cdot x - 0.08333333333333333 \cdot {x}^{4}}\right)} \]

      2. *-commutative 19.4%

         \[\leadsto 0.002777777777777778 \cdot {x}^{6} + \left(\frac{{x}^{4}}{x \cdot x - 0.08333333333333333 \cdot {x}^{4}} - \frac{\color{blue}{0.006944444444444444 \cdot {x}^{8}}}{x \cdot x - 0.08333333333333333 \cdot {x}^{4}}\right) \]

      3. metadata-eval 19.4%

         \[\leadsto 0.002777777777777778 \cdot {x}^{6} + \left(\frac{{x}^{4}}{x \cdot x - 0.08333333333333333 \cdot {x}^{4}} - \frac{0.006944444444444444 \cdot {x}^{\color{blue}{\left(2 \cdot 4\right)}}}{x \cdot x - 0.08333333333333333 \cdot {x}^{4}}\right) \]

      4. pow-sqr 19.4%

         \[\leadsto 0.002777777777777778 \cdot {x}^{6} + \left(\frac{{x}^{4}}{x \cdot x - 0.08333333333333333 \cdot {x}^{4}} - \frac{0.006944444444444444 \cdot \color{blue}{\left({x}^{4} \cdot {x}^{4}\right)}}{x \cdot x - 0.08333333333333333 \cdot {x}^{4}}\right) \]

      5. metadata-eval 19.4%

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

      6. swap-sqr 19.4%

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

      7. div-sub 19.4%

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

      8. sqr-pow 19.5%

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

      9. metadata-eval 19.5%

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

      10. pow2 19.5%

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

      11. metadata-eval 19.5%

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

      12. pow2 19.5%

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

      13. flip-+ 100.0%

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

      14. +-commutative 100.0%

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

      15. *-commutative 100.0%

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

   8. Applied egg-rr 100.0%

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

   if 3.9999999999999998e-7 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+r- 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+r- 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-+l- 100.0%

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

   3. Simplified 100.0%

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

      1. associate--r- 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-+r+ 100.0%

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

      6. cosh-undef 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 99.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (- (exp x) 2.0) (exp (- x))) 4e-7)
   (+ (* x x) (* 0.08333333333333333 (pow x 4.0)))
   (+ (* 2.0 (cosh x)) -2.0)))

C

double code(double x) {
	double tmp;
	if (((exp(x) - 2.0) + exp(-x)) <= 4e-7) {
		tmp = (x * x) + (0.08333333333333333 * pow(x, 4.0));
	} 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 (((exp(x) - 2.0d0) + exp(-x)) <= 4d-7) then
        tmp = (x * x) + (0.08333333333333333d0 * (x ** 4.0d0))
    else
        tmp = (2.0d0 * cosh(x)) + (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((Math.exp(x) - 2.0) + Math.exp(-x)) <= 4e-7) {
		tmp = (x * x) + (0.08333333333333333 * Math.pow(x, 4.0));
	} else {
		tmp = (2.0 * Math.cosh(x)) + -2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((math.exp(x) - 2.0) + math.exp(-x)) <= 4e-7:
		tmp = (x * x) + (0.08333333333333333 * math.pow(x, 4.0))
	else:
		tmp = (2.0 * math.cosh(x)) + -2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(exp(x) - 2.0) + exp(Float64(-x))) <= 4e-7)
		tmp = Float64(Float64(x * x) + Float64(0.08333333333333333 * (x ^ 4.0)));
	else
		tmp = Float64(Float64(2.0 * cosh(x)) + -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((exp(x) - 2.0) + exp(-x)) <= 4e-7)
		tmp = (x * x) + (0.08333333333333333 * (x ^ 4.0));
	else
		tmp = (2.0 * cosh(x)) + -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 4e-7], N[(N[(x * x), $MachinePrecision] + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 3.9999999999999998e-7

   1. Initial program 64.0%

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

      1. associate-+l- 64.0%

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

      2. sub-neg 64.0%

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

      3. sub-neg 64.0%

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

      4. +-commutative 64.0%

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

      5. distribute-neg-in 64.0%

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

      6. remove-double-neg 64.0%

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

      7. metadata-eval 64.0%

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

   3. Simplified 64.0%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. fma-def 100.0%

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

   6. Simplified 100.0%

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

      1. fma-udef 100.0%

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

   8. Applied egg-rr 100.0%

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

   if 3.9999999999999998e-7 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+r- 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+r- 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-+l- 100.0%

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

   3. Simplified 100.0%

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

      1. associate--r- 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-+r+ 100.0%

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

      6. cosh-undef 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 4: 87.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-5}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x + -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 8e-5) (* x x) (+ (* 2.0 (cosh x)) -2.0)))

C

double code(double x) {
	double tmp;
	if (x <= 8e-5) {
		tmp = x * x;
	} 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 <= 8d-5) then
        tmp = x * x
    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 <= 8e-5) {
		tmp = x * x;
	} else {
		tmp = (2.0 * Math.cosh(x)) + -2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 8e-5:
		tmp = x * x
	else:
		tmp = (2.0 * math.cosh(x)) + -2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 8e-5)
		tmp = Float64(x * x);
	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 <= 8e-5)
		tmp = x * x;
	else
		tmp = (2.0 * cosh(x)) + -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 8e-5], N[(x * x), $MachinePrecision], N[(N[(2.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.00000000000000065e-5

   1. Initial program 76.9%

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

      1. associate-+l- 76.8%

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

      2. sub-neg 76.8%

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

      3. sub-neg 76.8%

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

      4. +-commutative 76.8%

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

      5. distribute-neg-in 76.8%

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

      6. remove-double-neg 76.8%

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

      7. metadata-eval 76.8%

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

   3. Simplified 76.8%

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

   4. Taylor expanded in x around 0 81.9%

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

      1. unpow2 81.9%

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

   6. Simplified 81.9%

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

   if 8.00000000000000065e-5 < x

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-+r- 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r- 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+l- 99.5%

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

   3. Simplified 99.5%

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

      1. associate--r- 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

      4. +-commutative 99.6%

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

      5. associate-+r+ 99.5%

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

      6. cosh-undef 99.5%

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

   5. Applied egg-rr 99.5%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-5}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x + -2\\ \end{array} \]

Alternative 5: 87.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.6499999999999999

   1. Initial program 76.8%

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

      1. associate-+l- 76.8%

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

      2. sub-neg 76.8%

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

      3. sub-neg 76.8%

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

      4. +-commutative 76.8%

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

      5. distribute-neg-in 76.8%

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

      6. remove-double-neg 76.8%

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

      7. metadata-eval 76.8%

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

   3. Simplified 76.8%

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

   4. Taylor expanded in x around 0 81.8%

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

      1. unpow2 81.8%

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

   6. Simplified 81.8%

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

   if 1.6499999999999999 < x

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.1%

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

   5. Taylor expanded in x around inf 99.1%

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

      1. expm1-def 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 86.7%

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

Alternative 6: 75.4% accurate, 68.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.4%

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

   1. associate-+l- 83.4%

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

   2. sub-neg 83.4%

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

   3. sub-neg 83.4%

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

   4. +-commutative 83.4%

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

   5. distribute-neg-in 83.4%

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

   6. remove-double-neg 83.4%

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

   7. metadata-eval 83.4%

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

3. Simplified 83.4%

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

4. Taylor expanded in x around 0 73.3%

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

   1. unpow2 73.3%

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

6. Simplified 73.3%

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

7. Final simplification 73.3%

   \[\leadsto x \cdot x \]

Alternative 7: 3.7% accurate, 206.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

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

Wolfram

code[x_] := 2.0

Derivation

1. Initial program 83.4%

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

   1. associate-+l- 83.4%

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

   2. sub-neg 83.4%

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

   3. sub-neg 83.4%

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

   4. +-commutative 83.4%

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

   5. distribute-neg-in 83.4%

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

   6. remove-double-neg 83.4%

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

   7. metadata-eval 83.4%

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

3. Simplified 83.4%

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

   1. +-commutative 83.4%

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

   2. metadata-eval 83.4%

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

   3. sub-neg 83.4%

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

   4. associate--r- 83.4%

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

   5. add-sqr-sqrt 41.9%

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

   6. sqrt-unprod 82.6%

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

   7. sqr-neg 82.6%

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

   8. sqrt-unprod 40.7%

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

   9. add-sqr-sqrt 57.3%

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

5. Applied egg-rr 57.3%

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

   1. associate--r- 57.3%

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

   2. sub-neg 57.3%

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

   3. metadata-eval 57.3%

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

   4. +-commutative 57.3%

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

   5. rem-square-sqrt 27.9%

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

   6. fabs-sqr 27.9%

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

   7. rem-square-sqrt 30.5%

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

   8. metadata-eval 30.5%

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

   9. sub-neg 30.5%

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

   10. fabs-sub 30.5%

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

   11. rem-square-sqrt 2.6%

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

   12. fabs-sqr 2.6%

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

   13. rem-square-sqrt 2.7%

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

   14. remove-double-neg 2.7%

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

   15. remove-double-neg 2.7%

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

   16. distribute-neg-out 2.7%

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

   17. +-commutative 2.7%

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

   18. neg-sub0 2.7%

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

   19. associate--r- 2.7%

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

   20. metadata-eval 2.7%

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

7. Simplified 3.6%

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

8. Final simplification 3.6%

   \[\leadsto 2 \]

Alternative 8: 4.3% 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 83.4%

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

   1. associate-+l- 83.4%

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

   2. sub-neg 83.4%

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

   3. sub-neg 83.4%

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

   4. +-commutative 83.4%

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

   5. distribute-neg-in 83.4%

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

   6. remove-double-neg 83.4%

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

   7. metadata-eval 83.4%

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

3. Simplified 83.4%

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

4. Taylor expanded in x around 0 57.7%

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

5. Taylor expanded in x around 0 4.5%

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

6. Final simplification 4.5%

   \[\leadsto x \]

Developer target: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023297 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

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

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