NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.6% → 99.8%

Time: 15.0s

Alternatives: 12

Speedup: 9.7×

• 38.4% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -7.915983874990402e+262
  2. eps = -8.934309055095981e-96

Specification

Math

\[\begin{array}{l} \\ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (-
   (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x))))
   (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x)))))
  2.0))

C

double code(double x, double eps) {
	return (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (((1.0d0 + (1.0d0 / eps)) * exp(-((1.0d0 - eps) * x))) - (((1.0d0 / eps) - 1.0d0) * exp(-((1.0d0 + eps) * x)))) / 2.0d0
end function

Java

public static double code(double x, double eps) {
	return (((1.0 + (1.0 / eps)) * Math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * Math.exp(-((1.0 + eps) * x)))) / 2.0;
}

Python

def code(x, eps):
	return (((1.0 + (1.0 / eps)) * math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * math.exp(-((1.0 + eps) * x)))) / 2.0

Julia

function code(x, eps)
	return Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(-Float64(Float64(1.0 - eps) * x)))) - Float64(Float64(Float64(1.0 / eps) - 1.0) * exp(Float64(-Float64(Float64(1.0 + eps) * x))))) / 2.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[(1.0 - eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[(-N[(N[(1.0 + eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Initial Program: 73.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (-
   (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x))))
   (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x)))))
  2.0))

C

double code(double x, double eps) {
	return (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (((1.0d0 + (1.0d0 / eps)) * exp(-((1.0d0 - eps) * x))) - (((1.0d0 / eps) - 1.0d0) * exp(-((1.0d0 + eps) * x)))) / 2.0d0
end function

Java

public static double code(double x, double eps) {
	return (((1.0 + (1.0 / eps)) * Math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * Math.exp(-((1.0 + eps) * x)))) / 2.0;
}

Python

def code(x, eps):
	return (((1.0 + (1.0 / eps)) * math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * math.exp(-((1.0 + eps) * x)))) / 2.0

Julia

function code(x, eps)
	return Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(-Float64(Float64(1.0 - eps) * x)))) - Float64(Float64(Float64(1.0 / eps) - 1.0) * exp(Float64(-Float64(Float64(1.0 + eps) * x))))) / 2.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[(1.0 - eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[(-N[(N[(1.0 + eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x + 2, t\_0, x \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \cosh \left(x \cdot \varepsilon\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<=
        (+
         (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
         (* (exp (* x (- -1.0 eps))) (+ 1.0 (/ -1.0 eps))))
        0.0)
     (* 0.5 (fma (+ x 2.0) t_0 (* x t_0)))
     (* 0.5 (* 2.0 (cosh (* x eps)))))))

C

double code(double x, double eps) {
	double t_0 = exp(-x);
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 0.0) {
		tmp = 0.5 * fma((x + 2.0), t_0, (x * t_0));
	} else {
		tmp = 0.5 * (2.0 * cosh((x * eps)));
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(x * Float64(eps + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 + Float64(-1.0 / eps)))) <= 0.0)
		tmp = Float64(0.5 * fma(Float64(x + 2.0), t_0, Float64(x * t_0)));
	else
		tmp = Float64(0.5 * Float64(2.0 * cosh(Float64(x * eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[(N[(x + 2.0), $MachinePrecision] * t$95$0 + N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Cosh[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 0.0

   1. Initial program 31.5%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0

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

      1. *-lowering-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. distribute-rgt1-in N/A

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

      6. distribute-rgt-out-- N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. *-lowering-*.f64 N/A

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

      10. exp-lowering-exp.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. +-lowering-+.f64 N/A

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

   5. Simplified 99.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x} \cdot \left(\left(x + 2\right) + x\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(x + 2\right) \cdot e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right)} \]

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-lowering-+.f64 N/A

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

      4. exp-lowering-exp.f64 N/A

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

      5. neg-lowering-neg.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. exp-lowering-exp.f64 N/A

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

      8. neg-lowering-neg.f64 100.0

         \[\leadsto 0.5 \cdot \mathsf{fma}\left(x + 2, e^{-x}, x \cdot e^{\color{blue}{-x}}\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(x + 2, e^{-x}, x \cdot e^{-x}\right)} \]

   if 0.0 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-lowering-+.f64 N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. mul-1-neg N/A

          \[\leadsto \frac{1}{2} \cdot \left(e^{x \cdot \varepsilon} + e^{x \cdot \color{blue}{\left(\mathsf{neg}\left(\varepsilon\right)\right)}}\right) \]

       5. neg-lowering-neg.f64 100.0

          \[\leadsto 0.5 \cdot \left(e^{x \cdot \varepsilon} + e^{x \cdot \color{blue}{\left(-\varepsilon\right)}}\right) \]

   11. Simplified 100.0%

       \[\leadsto 0.5 \cdot \left(e^{x \cdot \varepsilon} + e^{\color{blue}{x \cdot \left(-\varepsilon\right)}}\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(e^{x \cdot \varepsilon} + e^{x \cdot \left(\mathsf{neg}\left(\varepsilon\right)\right)}\right) \cdot \frac{1}{2}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{x \cdot \varepsilon} + e^{x \cdot \left(\mathsf{neg}\left(\varepsilon\right)\right)}\right) \cdot \frac{1}{2}} \]

       3. distribute-rgt-neg-out N/A

          \[\leadsto \left(e^{x \cdot \varepsilon} + e^{\color{blue}{\mathsf{neg}\left(x \cdot \varepsilon\right)}}\right) \cdot \frac{1}{2} \]

       4. cosh-undef N/A

          \[\leadsto \color{blue}{\left(2 \cdot \cosh \left(x \cdot \varepsilon\right)\right)} \cdot \frac{1}{2} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(2 \cdot \cosh \left(x \cdot \varepsilon\right)\right)} \cdot \frac{1}{2} \]

       6. cosh-lowering-cosh.f64 N/A

          \[\leadsto \left(2 \cdot \color{blue}{\cosh \left(x \cdot \varepsilon\right)}\right) \cdot \frac{1}{2} \]

       7. *-lowering-*.f64 100.0

          \[\leadsto \left(2 \cdot \cosh \color{blue}{\left(x \cdot \varepsilon\right)}\right) \cdot 0.5 \]

   13. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\left(2 \cdot \cosh \left(x \cdot \varepsilon\right)\right) \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x + 2, e^{-x}, x \cdot e^{-x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \cosh \left(x \cdot \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(e^{-x} \cdot \left(x + \left(x + 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \cosh \left(x \cdot \varepsilon\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
       (* (exp (* x (- -1.0 eps))) (+ 1.0 (/ -1.0 eps))))
      0.0)
   (* 0.5 (* (exp (- x)) (+ x (+ x 2.0))))
   (* 0.5 (* 2.0 (cosh (* x eps))))))

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 0.0) {
		tmp = 0.5 * (exp(-x) * (x + (x + 2.0)));
	} else {
		tmp = 0.5 * (2.0 * cosh((x * eps)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((((1.0d0 + (1.0d0 / eps)) * exp((x * (eps + (-1.0d0))))) + (exp((x * ((-1.0d0) - eps))) * (1.0d0 + ((-1.0d0) / eps)))) <= 0.0d0) then
        tmp = 0.5d0 * (exp(-x) * (x + (x + 2.0d0)))
    else
        tmp = 0.5d0 * (2.0d0 * cosh((x * eps)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * Math.exp((x * (eps + -1.0)))) + (Math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 0.0) {
		tmp = 0.5 * (Math.exp(-x) * (x + (x + 2.0)));
	} else {
		tmp = 0.5 * (2.0 * Math.cosh((x * eps)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (((1.0 + (1.0 / eps)) * math.exp((x * (eps + -1.0)))) + (math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 0.0:
		tmp = 0.5 * (math.exp(-x) * (x + (x + 2.0)))
	else:
		tmp = 0.5 * (2.0 * math.cosh((x * eps)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(x * Float64(eps + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 + Float64(-1.0 / eps)))) <= 0.0)
		tmp = Float64(0.5 * Float64(exp(Float64(-x)) * Float64(x + Float64(x + 2.0))));
	else
		tmp = Float64(0.5 * Float64(2.0 * cosh(Float64(x * eps))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 0.0)
		tmp = 0.5 * (exp(-x) * (x + (x + 2.0)));
	else
		tmp = 0.5 * (2.0 * cosh((x * eps)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(0.5 * N[(N[Exp[(-x)], $MachinePrecision] * N[(x + N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(2.0 * N[Cosh[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 0.0

   1. Initial program 31.5%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0

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

      1. *-lowering-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. distribute-rgt1-in N/A

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

      6. distribute-rgt-out-- N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. *-lowering-*.f64 N/A

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

      10. exp-lowering-exp.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. +-lowering-+.f64 N/A

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

   5. Simplified 99.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x} \cdot \left(\left(x + 2\right) + x\right)\right)} \]

   if 0.0 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-lowering-+.f64 N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. mul-1-neg N/A

          \[\leadsto \frac{1}{2} \cdot \left(e^{x \cdot \varepsilon} + e^{x \cdot \color{blue}{\left(\mathsf{neg}\left(\varepsilon\right)\right)}}\right) \]

       5. neg-lowering-neg.f64 100.0

          \[\leadsto 0.5 \cdot \left(e^{x \cdot \varepsilon} + e^{x \cdot \color{blue}{\left(-\varepsilon\right)}}\right) \]

   11. Simplified 100.0%

       \[\leadsto 0.5 \cdot \left(e^{x \cdot \varepsilon} + e^{\color{blue}{x \cdot \left(-\varepsilon\right)}}\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(e^{x \cdot \varepsilon} + e^{x \cdot \left(\mathsf{neg}\left(\varepsilon\right)\right)}\right) \cdot \frac{1}{2}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{x \cdot \varepsilon} + e^{x \cdot \left(\mathsf{neg}\left(\varepsilon\right)\right)}\right) \cdot \frac{1}{2}} \]

       3. distribute-rgt-neg-out N/A

          \[\leadsto \left(e^{x \cdot \varepsilon} + e^{\color{blue}{\mathsf{neg}\left(x \cdot \varepsilon\right)}}\right) \cdot \frac{1}{2} \]

       4. cosh-undef N/A

          \[\leadsto \color{blue}{\left(2 \cdot \cosh \left(x \cdot \varepsilon\right)\right)} \cdot \frac{1}{2} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(2 \cdot \cosh \left(x \cdot \varepsilon\right)\right)} \cdot \frac{1}{2} \]

       6. cosh-lowering-cosh.f64 N/A

          \[\leadsto \left(2 \cdot \color{blue}{\cosh \left(x \cdot \varepsilon\right)}\right) \cdot \frac{1}{2} \]

       7. *-lowering-*.f64 100.0

          \[\leadsto \left(2 \cdot \cosh \color{blue}{\left(x \cdot \varepsilon\right)}\right) \cdot 0.5 \]

   13. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\left(2 \cdot \cosh \left(x \cdot \varepsilon\right)\right) \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 0:\\ \;\;\;\;0.5 \cdot \left(e^{-x} \cdot \left(x + \left(x + 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \cosh \left(x \cdot \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 0:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \cosh \left(x \cdot \varepsilon\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
       (* (exp (* x (- -1.0 eps))) (+ 1.0 (/ -1.0 eps))))
      0.0)
   (exp (- x))
   (* 0.5 (* 2.0 (cosh (* x eps))))))

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 0.0) {
		tmp = exp(-x);
	} else {
		tmp = 0.5 * (2.0 * cosh((x * eps)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((((1.0d0 + (1.0d0 / eps)) * exp((x * (eps + (-1.0d0))))) + (exp((x * ((-1.0d0) - eps))) * (1.0d0 + ((-1.0d0) / eps)))) <= 0.0d0) then
        tmp = exp(-x)
    else
        tmp = 0.5d0 * (2.0d0 * cosh((x * eps)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * Math.exp((x * (eps + -1.0)))) + (Math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 0.0) {
		tmp = Math.exp(-x);
	} else {
		tmp = 0.5 * (2.0 * Math.cosh((x * eps)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (((1.0 + (1.0 / eps)) * math.exp((x * (eps + -1.0)))) + (math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 0.0:
		tmp = math.exp(-x)
	else:
		tmp = 0.5 * (2.0 * math.cosh((x * eps)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(x * Float64(eps + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 + Float64(-1.0 / eps)))) <= 0.0)
		tmp = exp(Float64(-x));
	else
		tmp = Float64(0.5 * Float64(2.0 * cosh(Float64(x * eps))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 0.0)
		tmp = exp(-x);
	else
		tmp = 0.5 * (2.0 * cosh((x * eps)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[Exp[(-x)], $MachinePrecision], N[(0.5 * N[(2.0 * N[Cosh[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 0.0

   1. Initial program 31.5%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-lowering-+.f64 N/A

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

   5. Simplified 98.2%

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

   6. Taylor expanded in eps around 0

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

      1. distribute-rgt-in N/A

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

      2. neg-mul-1 N/A

         \[\leadsto e^{\mathsf{neg}\left(x\right)} \cdot \frac{1}{2} + e^{\color{blue}{\mathsf{neg}\left(x\right)}} \cdot \frac{1}{2} \]

      3. distribute-lft-out N/A

         \[\leadsto \color{blue}{e^{\mathsf{neg}\left(x\right)} \cdot \left(\frac{1}{2} + \frac{1}{2}\right)} \]

      4. metadata-eval N/A

         \[\leadsto e^{\mathsf{neg}\left(x\right)} \cdot \color{blue}{1} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{e^{\mathsf{neg}\left(x\right)} \cdot 1} \]

      6. neg-mul-1 N/A

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

      7. exp-lowering-exp.f64 N/A

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

      8. neg-mul-1 N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(x\right)}} \cdot 1 \]

      9. neg-lowering-neg.f64 98.2

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

   8. Simplified 98.2%

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

   if 0.0 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-lowering-+.f64 N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 N/A

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

       4. mul-1-neg N/A

          \[\leadsto \frac{1}{2} \cdot \left(e^{x \cdot \varepsilon} + e^{x \cdot \color{blue}{\left(\mathsf{neg}\left(\varepsilon\right)\right)}}\right) \]

       5. neg-lowering-neg.f64 100.0

          \[\leadsto 0.5 \cdot \left(e^{x \cdot \varepsilon} + e^{x \cdot \color{blue}{\left(-\varepsilon\right)}}\right) \]

   11. Simplified 100.0%

       \[\leadsto 0.5 \cdot \left(e^{x \cdot \varepsilon} + e^{\color{blue}{x \cdot \left(-\varepsilon\right)}}\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(e^{x \cdot \varepsilon} + e^{x \cdot \left(\mathsf{neg}\left(\varepsilon\right)\right)}\right) \cdot \frac{1}{2}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(e^{x \cdot \varepsilon} + e^{x \cdot \left(\mathsf{neg}\left(\varepsilon\right)\right)}\right) \cdot \frac{1}{2}} \]

       3. distribute-rgt-neg-out N/A

          \[\leadsto \left(e^{x \cdot \varepsilon} + e^{\color{blue}{\mathsf{neg}\left(x \cdot \varepsilon\right)}}\right) \cdot \frac{1}{2} \]

       4. cosh-undef N/A

          \[\leadsto \color{blue}{\left(2 \cdot \cosh \left(x \cdot \varepsilon\right)\right)} \cdot \frac{1}{2} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(2 \cdot \cosh \left(x \cdot \varepsilon\right)\right)} \cdot \frac{1}{2} \]

       6. cosh-lowering-cosh.f64 N/A

          \[\leadsto \left(2 \cdot \color{blue}{\cosh \left(x \cdot \varepsilon\right)}\right) \cdot \frac{1}{2} \]

       7. *-lowering-*.f64 100.0

          \[\leadsto \left(2 \cdot \cosh \color{blue}{\left(x \cdot \varepsilon\right)}\right) \cdot 0.5 \]

   13. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\left(2 \cdot \cosh \left(x \cdot \varepsilon\right)\right) \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 0:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 \cdot \cosh \left(x \cdot \varepsilon\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{1}{\varepsilon}\\ t_1 := -1 + \frac{1}{\varepsilon}\\ \mathbf{if}\;t\_0 \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 4:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{1}{\varepsilon} + \left(1 + t\_1\right)\right) \cdot \left(1 + \left(1 + \frac{0}{\varepsilon}\right)\right)}{t\_0 + t\_1}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ 1.0 eps))) (t_1 (+ -1.0 (/ 1.0 eps))))
   (if (<=
        (+
         (* t_0 (exp (* x (+ eps -1.0))))
         (* (exp (* x (- -1.0 eps))) (+ 1.0 (/ -1.0 eps))))
        4.0)
     (exp (- x))
     (/
      (/
       (* (+ (/ 1.0 eps) (+ 1.0 t_1)) (+ 1.0 (+ 1.0 (/ 0.0 eps))))
       (+ t_0 t_1))
      2.0))))

C

double code(double x, double eps) {
	double t_0 = 1.0 + (1.0 / eps);
	double t_1 = -1.0 + (1.0 / eps);
	double tmp;
	if (((t_0 * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 4.0) {
		tmp = exp(-x);
	} else {
		tmp = ((((1.0 / eps) + (1.0 + t_1)) * (1.0 + (1.0 + (0.0 / eps)))) / (t_0 + t_1)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (1.0d0 / eps)
    t_1 = (-1.0d0) + (1.0d0 / eps)
    if (((t_0 * exp((x * (eps + (-1.0d0))))) + (exp((x * ((-1.0d0) - eps))) * (1.0d0 + ((-1.0d0) / eps)))) <= 4.0d0) then
        tmp = exp(-x)
    else
        tmp = ((((1.0d0 / eps) + (1.0d0 + t_1)) * (1.0d0 + (1.0d0 + (0.0d0 / eps)))) / (t_0 + t_1)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 1.0 + (1.0 / eps);
	double t_1 = -1.0 + (1.0 / eps);
	double tmp;
	if (((t_0 * Math.exp((x * (eps + -1.0)))) + (Math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 4.0) {
		tmp = Math.exp(-x);
	} else {
		tmp = ((((1.0 / eps) + (1.0 + t_1)) * (1.0 + (1.0 + (0.0 / eps)))) / (t_0 + t_1)) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 1.0 + (1.0 / eps)
	t_1 = -1.0 + (1.0 / eps)
	tmp = 0
	if ((t_0 * math.exp((x * (eps + -1.0)))) + (math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 4.0:
		tmp = math.exp(-x)
	else:
		tmp = ((((1.0 / eps) + (1.0 + t_1)) * (1.0 + (1.0 + (0.0 / eps)))) / (t_0 + t_1)) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(1.0 + Float64(1.0 / eps))
	t_1 = Float64(-1.0 + Float64(1.0 / eps))
	tmp = 0.0
	if (Float64(Float64(t_0 * exp(Float64(x * Float64(eps + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 + Float64(-1.0 / eps)))) <= 4.0)
		tmp = exp(Float64(-x));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 / eps) + Float64(1.0 + t_1)) * Float64(1.0 + Float64(1.0 + Float64(0.0 / eps)))) / Float64(t_0 + t_1)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = 1.0 + (1.0 / eps);
	t_1 = -1.0 + (1.0 / eps);
	tmp = 0.0;
	if (((t_0 * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 4.0)
		tmp = exp(-x);
	else
		tmp = ((((1.0 / eps) + (1.0 + t_1)) * (1.0 + (1.0 + (0.0 / eps)))) / (t_0 + t_1)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0], N[Exp[(-x)], $MachinePrecision], N[(N[(N[(N[(N[(1.0 / eps), $MachinePrecision] + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[(0.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 4

   1. Initial program 51.6%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-lowering-+.f64 N/A

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

   5. Simplified 98.7%

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

   6. Taylor expanded in eps around 0

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

      1. distribute-rgt-in N/A

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

      2. neg-mul-1 N/A

         \[\leadsto e^{\mathsf{neg}\left(x\right)} \cdot \frac{1}{2} + e^{\color{blue}{\mathsf{neg}\left(x\right)}} \cdot \frac{1}{2} \]

      3. distribute-lft-out N/A

         \[\leadsto \color{blue}{e^{\mathsf{neg}\left(x\right)} \cdot \left(\frac{1}{2} + \frac{1}{2}\right)} \]

      4. metadata-eval N/A

         \[\leadsto e^{\mathsf{neg}\left(x\right)} \cdot \color{blue}{1} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{e^{\mathsf{neg}\left(x\right)} \cdot 1} \]

      6. neg-mul-1 N/A

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

      7. exp-lowering-exp.f64 N/A

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

      8. neg-mul-1 N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(x\right)}} \cdot 1 \]

      9. neg-lowering-neg.f64 98.2

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

   8. Simplified 98.2%

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

   if 4 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\mathsf{neg}\left(\left(1 - \varepsilon\right) \cdot x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 N/A

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

      5. /-lowering-/.f64 54.3

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

   5. Simplified 54.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(-1 + \frac{1}{\varepsilon}\right)}{2} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(-1 + \frac{1}{\varepsilon}\right)}{2} \]

      2. /-lowering-/.f64 3.1

         \[\leadsto \frac{\left(1 + \color{blue}{\frac{1}{\varepsilon}}\right) - \left(-1 + \frac{1}{\varepsilon}\right)}{2} \]

   8. Simplified 3.1%

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(-1 + \frac{1}{\varepsilon}\right)}{2} \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) + \left(\mathsf{neg}\left(\left(-1 + \frac{1}{\varepsilon}\right)\right)\right)}}{2} \]

      2. flip-+ N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) - \left(\mathsf{neg}\left(\left(-1 + \frac{1}{\varepsilon}\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(-1 + \frac{1}{\varepsilon}\right)\right)\right)}{\left(1 + \frac{1}{\varepsilon}\right) - \left(\mathsf{neg}\left(\left(-1 + \frac{1}{\varepsilon}\right)\right)\right)}}}{2} \]

      3. sqr-neg N/A

         \[\leadsto \frac{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) - \color{blue}{\left(-1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)}}{\left(1 + \frac{1}{\varepsilon}\right) - \left(\mathsf{neg}\left(\left(-1 + \frac{1}{\varepsilon}\right)\right)\right)}}{2} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) - \left(-1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)}{\left(1 + \frac{1}{\varepsilon}\right) - \left(\mathsf{neg}\left(\left(-1 + \frac{1}{\varepsilon}\right)\right)\right)}}}{2} \]

   10. Applied egg-rr 49.1%

       \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{\varepsilon} + \left(1 + \left(-1 + \frac{1}{\varepsilon}\right)\right)\right) \cdot \left(1 + \left(\frac{0}{\varepsilon} - -1\right)\right)}{\left(1 + \frac{1}{\varepsilon}\right) - \left(1 - \frac{1}{\varepsilon}\right)}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 4:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{1}{\varepsilon} + \left(1 + \left(-1 + \frac{1}{\varepsilon}\right)\right)\right) \cdot \left(1 + \left(1 + \frac{0}{\varepsilon}\right)\right)}{\left(1 + \frac{1}{\varepsilon}\right) + \left(-1 + \frac{1}{\varepsilon}\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{1}{\varepsilon}\\ t_1 := -1 + \frac{1}{\varepsilon}\\ \mathbf{if}\;t\_0 \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 4:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left(x \cdot \varepsilon\right) \cdot \left(x \cdot \varepsilon\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{1}{\varepsilon} + \left(1 + t\_1\right)\right) \cdot \left(1 + \left(1 + \frac{0}{\varepsilon}\right)\right)}{t\_0 + t\_1}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ 1.0 eps))) (t_1 (+ -1.0 (/ 1.0 eps))))
   (if (<=
        (+
         (* t_0 (exp (* x (+ eps -1.0))))
         (* (exp (* x (- -1.0 eps))) (+ 1.0 (/ -1.0 eps))))
        4.0)
     (fma 0.5 (* (* x eps) (* x eps)) 1.0)
     (/
      (/
       (* (+ (/ 1.0 eps) (+ 1.0 t_1)) (+ 1.0 (+ 1.0 (/ 0.0 eps))))
       (+ t_0 t_1))
      2.0))))

C

double code(double x, double eps) {
	double t_0 = 1.0 + (1.0 / eps);
	double t_1 = -1.0 + (1.0 / eps);
	double tmp;
	if (((t_0 * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 4.0) {
		tmp = fma(0.5, ((x * eps) * (x * eps)), 1.0);
	} else {
		tmp = ((((1.0 / eps) + (1.0 + t_1)) * (1.0 + (1.0 + (0.0 / eps)))) / (t_0 + t_1)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 + Float64(1.0 / eps))
	t_1 = Float64(-1.0 + Float64(1.0 / eps))
	tmp = 0.0
	if (Float64(Float64(t_0 * exp(Float64(x * Float64(eps + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 + Float64(-1.0 / eps)))) <= 4.0)
		tmp = fma(0.5, Float64(Float64(x * eps) * Float64(x * eps)), 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 / eps) + Float64(1.0 + t_1)) * Float64(1.0 + Float64(1.0 + Float64(0.0 / eps)))) / Float64(t_0 + t_1)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0], N[(0.5 * N[(N[(x * eps), $MachinePrecision] * N[(x * eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(N[(N[(1.0 / eps), $MachinePrecision] + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + N[(0.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 4

   1. Initial program 51.6%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 75.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x \cdot \mathsf{fma}\left(\varepsilon + 1, -1 + \frac{1}{\varepsilon}, \mathsf{fma}\left(1 + \frac{1}{\varepsilon}, \mathsf{fma}\left(0.5 \cdot x, \left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right), -1 + \varepsilon\right), -\left(\mathsf{fma}\left(x, \varepsilon, x\right) \cdot \left(\varepsilon + 1\right)\right) \cdot \left(\left(-1 + \frac{1}{\varepsilon}\right) \cdot 0.5\right)\right)\right), 1\right)} \]

   6. Taylor expanded in eps around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{2} \cdot x - \frac{-1}{2} \cdot x\right)\right)}, 1\right) \]
   7. Step-by-step derivation

      1. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x \cdot \left({\varepsilon}^{2} \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} - \frac{-1}{2}\right)\right)}\right), 1\right) \]

      2. metadata-eval N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right), 1\right) \]

      7. *-lowering-*.f64 76.6

         \[\leadsto \mathsf{fma}\left(0.5, x \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right), 1\right) \]

   8. Simplified 76.6%

      \[\leadsto \mathsf{fma}\left(0.5, x \cdot \color{blue}{\left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, 1\right) \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \varepsilon\right)}, 1\right) \]

      2. unswap-sqr N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(x \cdot \varepsilon\right) \cdot \left(x \cdot \varepsilon\right)}, 1\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(x \cdot \varepsilon\right) \cdot \left(x \cdot \varepsilon\right)}, 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\left(x \cdot \varepsilon\right)} \cdot \left(x \cdot \varepsilon\right), 1\right) \]

      5. *-lowering-*.f64 79.2

         \[\leadsto \mathsf{fma}\left(0.5, \left(x \cdot \varepsilon\right) \cdot \color{blue}{\left(x \cdot \varepsilon\right)}, 1\right) \]

   10. Applied egg-rr 79.2%

       \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\left(x \cdot \varepsilon\right) \cdot \left(x \cdot \varepsilon\right)}, 1\right) \]

   if 4 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\mathsf{neg}\left(\left(1 - \varepsilon\right) \cdot x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 N/A

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

      5. /-lowering-/.f64 54.3

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

   5. Simplified 54.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(-1 + \frac{1}{\varepsilon}\right)}{2} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(-1 + \frac{1}{\varepsilon}\right)}{2} \]

      2. /-lowering-/.f64 3.1

         \[\leadsto \frac{\left(1 + \color{blue}{\frac{1}{\varepsilon}}\right) - \left(-1 + \frac{1}{\varepsilon}\right)}{2} \]

   8. Simplified 3.1%

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(-1 + \frac{1}{\varepsilon}\right)}{2} \]
   9. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) + \left(\mathsf{neg}\left(\left(-1 + \frac{1}{\varepsilon}\right)\right)\right)}}{2} \]

      2. flip-+ N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) - \left(\mathsf{neg}\left(\left(-1 + \frac{1}{\varepsilon}\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(-1 + \frac{1}{\varepsilon}\right)\right)\right)}{\left(1 + \frac{1}{\varepsilon}\right) - \left(\mathsf{neg}\left(\left(-1 + \frac{1}{\varepsilon}\right)\right)\right)}}}{2} \]

      3. sqr-neg N/A

         \[\leadsto \frac{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) - \color{blue}{\left(-1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)}}{\left(1 + \frac{1}{\varepsilon}\right) - \left(\mathsf{neg}\left(\left(-1 + \frac{1}{\varepsilon}\right)\right)\right)}}{2} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) - \left(-1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)}{\left(1 + \frac{1}{\varepsilon}\right) - \left(\mathsf{neg}\left(\left(-1 + \frac{1}{\varepsilon}\right)\right)\right)}}}{2} \]

   10. Applied egg-rr 49.1%

       \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{\varepsilon} + \left(1 + \left(-1 + \frac{1}{\varepsilon}\right)\right)\right) \cdot \left(1 + \left(\frac{0}{\varepsilon} - -1\right)\right)}{\left(1 + \frac{1}{\varepsilon}\right) - \left(1 - \frac{1}{\varepsilon}\right)}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 4:\\ \;\;\;\;\mathsf{fma}\left(0.5, \left(x \cdot \varepsilon\right) \cdot \left(x \cdot \varepsilon\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{1}{\varepsilon} + \left(1 + \left(-1 + \frac{1}{\varepsilon}\right)\right)\right) \cdot \left(1 + \left(1 + \frac{0}{\varepsilon}\right)\right)}{\left(1 + \frac{1}{\varepsilon}\right) + \left(-1 + \frac{1}{\varepsilon}\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (+
       (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
       (* (exp (* x (- -1.0 eps))) (+ 1.0 (/ -1.0 eps))))
      4.0)
   1.0
   (* 0.5 (* x (* x (* eps eps))))))

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 4.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (x * (x * (eps * eps)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((((1.0d0 + (1.0d0 / eps)) * exp((x * (eps + (-1.0d0))))) + (exp((x * ((-1.0d0) - eps))) * (1.0d0 + ((-1.0d0) / eps)))) <= 4.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 * (x * (x * (eps * eps)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((((1.0 + (1.0 / eps)) * Math.exp((x * (eps + -1.0)))) + (Math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 4.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (x * (x * (eps * eps)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (((1.0 + (1.0 / eps)) * math.exp((x * (eps + -1.0)))) + (math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 4.0:
		tmp = 1.0
	else:
		tmp = 0.5 * (x * (x * (eps * eps)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(x * Float64(eps + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 + Float64(-1.0 / eps)))) <= 4.0)
		tmp = 1.0;
	else
		tmp = Float64(0.5 * Float64(x * Float64(x * Float64(eps * eps))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) <= 4.0)
		tmp = 1.0;
	else
		tmp = 0.5 * (x * (x * (eps * eps)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0], 1.0, N[(0.5 * N[(x * N[(x * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 4

   1. Initial program 51.6%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 79.0%

      \[\leadsto \color{blue}{1} \]

   if 4 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 86.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x \cdot \mathsf{fma}\left(\varepsilon + 1, -1 + \frac{1}{\varepsilon}, \mathsf{fma}\left(1 + \frac{1}{\varepsilon}, \mathsf{fma}\left(0.5 \cdot x, \left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right), -1 + \varepsilon\right), -\left(\mathsf{fma}\left(x, \varepsilon, x\right) \cdot \left(\varepsilon + 1\right)\right) \cdot \left(\left(-1 + \frac{1}{\varepsilon}\right) \cdot 0.5\right)\right)\right), 1\right)} \]

   6. Taylor expanded in eps around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{2} \cdot x - \frac{-1}{2} \cdot x\right)\right)}, 1\right) \]
   7. Step-by-step derivation

      1. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x \cdot \left({\varepsilon}^{2} \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} - \frac{-1}{2}\right)\right)}\right), 1\right) \]

      2. metadata-eval N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right), 1\right) \]

      7. *-lowering-*.f64 86.6

         \[\leadsto \mathsf{fma}\left(0.5, x \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right), 1\right) \]

   8. Simplified 86.6%

      \[\leadsto \mathsf{fma}\left(0.5, x \cdot \color{blue}{\left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, 1\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {x}^{2}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 N/A

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

       9. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right) \]

       10. *-lowering-*.f64 86.6

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

   11. Simplified 86.6%

       \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.0% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (* 0.5 (+ (exp (- (* x eps) x)) (exp (* x (- -1.0 eps))))))

C

double code(double x, double eps) {
	return 0.5 * (exp(((x * eps) - x)) + exp((x * (-1.0 - eps))));
}

Fortran

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

Java

public static double code(double x, double eps) {
	return 0.5 * (Math.exp(((x * eps) - x)) + Math.exp((x * (-1.0 - eps))));
}

Python

def code(x, eps):
	return 0.5 * (math.exp(((x * eps) - x)) + math.exp((x * (-1.0 - eps))))

Julia

function code(x, eps)
	return Float64(0.5 * Float64(exp(Float64(Float64(x * eps) - x)) + exp(Float64(x * Float64(-1.0 - eps)))))
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.5 * (exp(((x * eps) - x)) + exp((x * (-1.0 - eps))));
end

Wolfram

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

Derivation

1. Initial program 71.6%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
2. Add Preprocessing

3. Taylor expanded in eps around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. +-lowering-+.f64 N/A

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

5. Simplified 99.2%

   \[\leadsto \color{blue}{0.5 \cdot \left(e^{x \cdot \varepsilon - x} + e^{x \cdot \left(-1 - \varepsilon\right)}\right)} \]
6. Add Preprocessing

Alternative 8: 83.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0078:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \mathsf{fma}\left(\varepsilon + 1, -1 + \frac{1}{\varepsilon}, \frac{\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \left(1 - x\right) + \mathsf{fma}\left(x, \varepsilon, x\right), 0\right) - x, -1\right)}{\varepsilon}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 0.0078)
   (fma
    0.5
    (*
     x
     (fma
      (+ eps 1.0)
      (+ -1.0 (/ 1.0 eps))
      (/ (fma eps (- (fma eps (+ (- 1.0 x) (fma x eps x)) 0.0) x) -1.0) eps)))
    1.0)
   (* 0.5 (* x (* x (* eps eps))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 0.0078) {
		tmp = fma(0.5, (x * fma((eps + 1.0), (-1.0 + (1.0 / eps)), (fma(eps, (fma(eps, ((1.0 - x) + fma(x, eps, x)), 0.0) - x), -1.0) / eps))), 1.0);
	} else {
		tmp = 0.5 * (x * (x * (eps * eps)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 0.0078)
		tmp = fma(0.5, Float64(x * fma(Float64(eps + 1.0), Float64(-1.0 + Float64(1.0 / eps)), Float64(fma(eps, Float64(fma(eps, Float64(Float64(1.0 - x) + fma(x, eps, x)), 0.0) - x), -1.0) / eps))), 1.0);
	else
		tmp = Float64(0.5 * Float64(x * Float64(x * Float64(eps * eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 0.0078], N[(0.5 * N[(x * N[(N[(eps + 1.0), $MachinePrecision] * N[(-1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] + N[(N[(eps * N[(N[(eps * N[(N[(1.0 - x), $MachinePrecision] + N[(x * eps + x), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision] - x), $MachinePrecision] + -1.0), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(0.5 * N[(x * N[(x * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0077999999999999996

   1. Initial program 61.4%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 91.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x \cdot \mathsf{fma}\left(\varepsilon + 1, -1 + \frac{1}{\varepsilon}, \mathsf{fma}\left(1 + \frac{1}{\varepsilon}, \mathsf{fma}\left(0.5 \cdot x, \left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right), -1 + \varepsilon\right), -\left(\mathsf{fma}\left(x, \varepsilon, x\right) \cdot \left(\varepsilon + 1\right)\right) \cdot \left(\left(-1 + \frac{1}{\varepsilon}\right) \cdot 0.5\right)\right)\right), 1\right)} \]

   6. Taylor expanded in eps around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x \cdot \mathsf{fma}\left(\varepsilon + 1, -1 + \frac{1}{\varepsilon}, \color{blue}{\frac{\varepsilon \cdot \left(\left(-1 \cdot x + \left(\frac{1}{2} \cdot x + \varepsilon \cdot \left(\left(1 + \left(-1 \cdot x + \left(\frac{1}{2} \cdot x + \varepsilon \cdot \left(\frac{1}{2} \cdot x - \frac{-1}{2} \cdot x\right)\right)\right)\right) - \frac{1}{2} \cdot \left(x + -2 \cdot x\right)\right)\right)\right) - \frac{1}{2} \cdot \left(-1 \cdot x + 2 \cdot x\right)\right) - 1}{\varepsilon}}\right), 1\right) \]

   8. Simplified 93.9%

      \[\leadsto \mathsf{fma}\left(0.5, x \cdot \mathsf{fma}\left(\varepsilon + 1, -1 + \frac{1}{\varepsilon}, \color{blue}{\frac{\mathsf{fma}\left(\varepsilon, \left(-x\right) + \mathsf{fma}\left(\varepsilon, \left(1 - x\right) + \mathsf{fma}\left(x, \varepsilon, x\right), 0\right), -1\right)}{\varepsilon}}\right), 1\right) \]

   if 0.0077999999999999996 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 46.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x \cdot \mathsf{fma}\left(\varepsilon + 1, -1 + \frac{1}{\varepsilon}, \mathsf{fma}\left(1 + \frac{1}{\varepsilon}, \mathsf{fma}\left(0.5 \cdot x, \left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right), -1 + \varepsilon\right), -\left(\mathsf{fma}\left(x, \varepsilon, x\right) \cdot \left(\varepsilon + 1\right)\right) \cdot \left(\left(-1 + \frac{1}{\varepsilon}\right) \cdot 0.5\right)\right)\right), 1\right)} \]

   6. Taylor expanded in eps around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{2} \cdot x - \frac{-1}{2} \cdot x\right)\right)}, 1\right) \]
   7. Step-by-step derivation

      1. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x \cdot \left({\varepsilon}^{2} \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} - \frac{-1}{2}\right)\right)}\right), 1\right) \]

      2. metadata-eval N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right), 1\right) \]

      7. *-lowering-*.f64 47.3

         \[\leadsto \mathsf{fma}\left(0.5, x \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right), 1\right) \]

   8. Simplified 47.3%

      \[\leadsto \mathsf{fma}\left(0.5, x \cdot \color{blue}{\left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, 1\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {x}^{2}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 N/A

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

       9. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right) \]

       10. *-lowering-*.f64 64.4

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

   11. Simplified 64.4%

       \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0078:\\ \;\;\;\;\mathsf{fma}\left(0.5, x \cdot \mathsf{fma}\left(\varepsilon + 1, -1 + \frac{1}{\varepsilon}, \frac{\mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\varepsilon, \left(1 - x\right) + \mathsf{fma}\left(x, \varepsilon, x\right), 0\right) - x, -1\right)}{\varepsilon}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.0% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{if}\;x \leq 0.0078:\\ \;\;\;\;\mathsf{fma}\left(0.5, t\_0, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (* x (* eps eps)))))
   (if (<= x 0.0078) (fma 0.5 t_0 1.0) (* 0.5 t_0))))

C

double code(double x, double eps) {
	double t_0 = x * (x * (eps * eps));
	double tmp;
	if (x <= 0.0078) {
		tmp = fma(0.5, t_0, 1.0);
	} else {
		tmp = 0.5 * t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x * Float64(eps * eps)))
	tmp = 0.0
	if (x <= 0.0078)
		tmp = fma(0.5, t_0, 1.0);
	else
		tmp = Float64(0.5 * t_0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.0078], N[(0.5 * t$95$0 + 1.0), $MachinePrecision], N[(0.5 * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0077999999999999996

   1. Initial program 61.4%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 91.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x \cdot \mathsf{fma}\left(\varepsilon + 1, -1 + \frac{1}{\varepsilon}, \mathsf{fma}\left(1 + \frac{1}{\varepsilon}, \mathsf{fma}\left(0.5 \cdot x, \left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right), -1 + \varepsilon\right), -\left(\mathsf{fma}\left(x, \varepsilon, x\right) \cdot \left(\varepsilon + 1\right)\right) \cdot \left(\left(-1 + \frac{1}{\varepsilon}\right) \cdot 0.5\right)\right)\right), 1\right)} \]

   6. Taylor expanded in eps around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{2} \cdot x - \frac{-1}{2} \cdot x\right)\right)}, 1\right) \]
   7. Step-by-step derivation

      1. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x \cdot \left({\varepsilon}^{2} \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} - \frac{-1}{2}\right)\right)}\right), 1\right) \]

      2. metadata-eval N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right), 1\right) \]

      7. *-lowering-*.f64 92.9

         \[\leadsto \mathsf{fma}\left(0.5, x \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right), 1\right) \]

   8. Simplified 92.9%

      \[\leadsto \mathsf{fma}\left(0.5, x \cdot \color{blue}{\left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, 1\right) \]

   if 0.0077999999999999996 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Simplified 46.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, x \cdot \mathsf{fma}\left(\varepsilon + 1, -1 + \frac{1}{\varepsilon}, \mathsf{fma}\left(1 + \frac{1}{\varepsilon}, \mathsf{fma}\left(0.5 \cdot x, \left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right), -1 + \varepsilon\right), -\left(\mathsf{fma}\left(x, \varepsilon, x\right) \cdot \left(\varepsilon + 1\right)\right) \cdot \left(\left(-1 + \frac{1}{\varepsilon}\right) \cdot 0.5\right)\right)\right), 1\right)} \]

   6. Taylor expanded in eps around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{2} \cdot x - \frac{-1}{2} \cdot x\right)\right)}, 1\right) \]
   7. Step-by-step derivation

      1. distribute-rgt-out-- N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x \cdot \left({\varepsilon}^{2} \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} - \frac{-1}{2}\right)\right)}\right), 1\right) \]

      2. metadata-eval N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right), 1\right) \]

      7. *-lowering-*.f64 47.3

         \[\leadsto \mathsf{fma}\left(0.5, x \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right), 1\right) \]

   8. Simplified 47.3%

      \[\leadsto \mathsf{fma}\left(0.5, x \cdot \color{blue}{\left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}, 1\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {x}^{2}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 N/A

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

       7. *-commutative N/A

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

       8. *-lowering-*.f64 N/A

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

       9. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right)\right) \]

       10. *-lowering-*.f64 64.4

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

   11. Simplified 64.4%

       \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 63.0% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -430:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -430.0)
   (fma -0.16666666666666666 (* x (* x x)) 1.0)
   (fma (* x x) (fma x 0.3333333333333333 -0.5) 1.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -430.0) {
		tmp = fma(-0.16666666666666666, (x * (x * x)), 1.0);
	} else {
		tmp = fma((x * x), fma(x, 0.3333333333333333, -0.5), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -430.0)
		tmp = fma(-0.16666666666666666, Float64(x * Float64(x * x)), 1.0);
	else
		tmp = fma(Float64(x * x), fma(x, 0.3333333333333333, -0.5), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -430.0], N[(-0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * 0.3333333333333333 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -430

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-lowering-+.f64 N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(2 + x \cdot \left(\left(\varepsilon + \left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2} + \left(\frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2} + x \cdot \left(\frac{-1}{6} \cdot {\left(1 + \varepsilon\right)}^{3} + \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right)\right)\right)\right)\right) - 1\right)\right)} \]

   8. Simplified 16.7%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(x, \varepsilon + \mathsf{fma}\left(x, \mathsf{fma}\left(0.5, \mathsf{fma}\left(1 + \varepsilon, 1 + \varepsilon, \left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right), x \cdot \mathsf{fma}\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right), \left(\varepsilon + -1\right) \cdot 0.16666666666666666, \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot -0.16666666666666666\right)\right)\right), -1 + \left(-1 - \varepsilon\right)\right), 2\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot {\left(1 + \varepsilon\right)}^{3} + \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right)}, 2\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot {\left(1 + \varepsilon\right)}^{3} + \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right)}, 2\right) \]

       2. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-1}{6} \cdot {\left(1 + \varepsilon\right)}^{3} + \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right), 2\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-1}{6} \cdot {\left(1 + \varepsilon\right)}^{3} + \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right), 2\right) \]

       4. cube-mult N/A

          \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\left(1 + \varepsilon\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)} + \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right), 2\right) \]

       5. unpow2 N/A

          \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \left(\frac{-1}{6} \cdot \left(\left(1 + \varepsilon\right) \cdot \color{blue}{{\left(1 + \varepsilon\right)}^{2}}\right) + \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right), 2\right) \]

       6. associate-*r* N/A

          \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot \left(1 + \varepsilon\right)\right) \cdot {\left(1 + \varepsilon\right)}^{2}} + \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right), 2\right) \]

       7. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6} \cdot \left(1 + \varepsilon\right), {\left(1 + \varepsilon\right)}^{2}, \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right)}, 2\right) \]

       8. +-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot \color{blue}{\left(\varepsilon + 1\right)}, {\left(1 + \varepsilon\right)}^{2}, \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right), 2\right) \]

       9. distribute-rgt-in N/A

          \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \frac{-1}{6} + 1 \cdot \frac{-1}{6}}, {\left(1 + \varepsilon\right)}^{2}, \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right), 2\right) \]

       10. metadata-eval N/A

           \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \frac{-1}{6} + \color{blue}{\frac{-1}{6}}, {\left(1 + \varepsilon\right)}^{2}, \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right), 2\right) \]

       11. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\varepsilon, \frac{-1}{6}, \frac{-1}{6}\right)}, {\left(1 + \varepsilon\right)}^{2}, \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right), 2\right) \]

       12. unpow2 N/A

           \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \frac{-1}{6}, \frac{-1}{6}\right), \color{blue}{\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)}, \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right), 2\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \frac{-1}{6}, \frac{-1}{6}\right), \color{blue}{\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)}, \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right), 2\right) \]

       14. +-lowering-+.f64 N/A

           \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \frac{-1}{6}, \frac{-1}{6}\right), \color{blue}{\left(1 + \varepsilon\right)} \cdot \left(1 + \varepsilon\right), \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right), 2\right) \]

       15. +-lowering-+.f64 N/A

           \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \frac{-1}{6}, \frac{-1}{6}\right), \left(1 + \varepsilon\right) \cdot \color{blue}{\left(1 + \varepsilon\right)}, \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right), 2\right) \]

       16. cube-mult N/A

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

       17. unpow2 N/A

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

       18. associate-*r* N/A

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

       19. *-lowering-*.f64 N/A

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

   11. Simplified 22.4%

       \[\leadsto 0.5 \cdot \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, -0.16666666666666666, -0.16666666666666666\right), \left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right), \mathsf{fma}\left(\varepsilon, 0.16666666666666666, -0.16666666666666666\right) \cdot \left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right)\right)}, 2\right) \]

   12. Taylor expanded in eps around 0

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(2 + \frac{-1}{3} \cdot {x}^{3}\right)} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{-1}{3} \cdot {x}^{3} + 2\right)} \]

       2. distribute-lft-in N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\frac{-1}{3} \cdot {x}^{3}\right) + \frac{1}{2} \cdot 2} \]

       3. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{-1}{3}\right) \cdot {x}^{3}} + \frac{1}{2} \cdot 2 \]

       4. metadata-eval N/A

          \[\leadsto \color{blue}{\frac{-1}{6}} \cdot {x}^{3} + \frac{1}{2} \cdot 2 \]

       5. metadata-eval N/A

          \[\leadsto \frac{-1}{6} \cdot {x}^{3} + \color{blue}{1} \]

       6. accelerator-lowering-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {x}^{3}, 1\right)} \]

       7. cube-mult N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{x \cdot \left(x \cdot x\right)}, 1\right) \]

       8. unpow2 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \color{blue}{\left(x \cdot x\right)}, 1\right) \]

       11. *-lowering-*.f64 71.2

           \[\leadsto \mathsf{fma}\left(-0.16666666666666666, x \cdot \color{blue}{\left(x \cdot x\right)}, 1\right) \]

   14. Simplified 71.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), 1\right)} \]

   if -430 < x

   1. Initial program 67.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0

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

      1. *-lowering-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. associate-+l- N/A

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

      5. distribute-rgt1-in N/A

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

      6. distribute-rgt-out-- N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. *-lowering-*.f64 N/A

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

      10. exp-lowering-exp.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. +-lowering-+.f64 N/A

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

   5. Simplified 68.6%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x} \cdot \left(\left(x + 2\right) + x\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + 1} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{3} \cdot x - \frac{1}{2}, 1\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{3} \cdot x - \frac{1}{2}, 1\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{3} \cdot x - \frac{1}{2}, 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{3} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \frac{1}{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \frac{1}{3} + \color{blue}{\frac{-1}{2}}, 1\right) \]

      8. accelerator-lowering-fma.f64 65.4

         \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, 0.3333333333333333, -0.5\right)}, 1\right) \]

   8. Simplified 65.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, 0.3333333333333333, -0.5\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 52.7% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), 1\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma -0.16666666666666666 (* x (* x x)) 1.0))

C

double code(double x, double eps) {
	return fma(-0.16666666666666666, (x * (x * x)), 1.0);
}

Julia

function code(x, eps)
	return fma(-0.16666666666666666, Float64(x * Float64(x * x)), 1.0)
end

Wolfram

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

Derivation

1. Initial program 71.6%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
2. Add Preprocessing

3. Taylor expanded in eps around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. +-lowering-+.f64 N/A

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

5. Simplified 99.2%

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

6. Taylor expanded in x around 0

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(2 + x \cdot \left(\left(\varepsilon + \left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2} + \left(\frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2} + x \cdot \left(\frac{-1}{6} \cdot {\left(1 + \varepsilon\right)}^{3} + \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right)\right)\right)\right)\right) - 1\right)\right)} \]

8. Simplified 45.8%

   \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(x, \varepsilon + \mathsf{fma}\left(x, \mathsf{fma}\left(0.5, \mathsf{fma}\left(1 + \varepsilon, 1 + \varepsilon, \left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right), x \cdot \mathsf{fma}\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right), \left(\varepsilon + -1\right) \cdot 0.16666666666666666, \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot -0.16666666666666666\right)\right)\right), -1 + \left(-1 - \varepsilon\right)\right), 2\right)} \]

9. Taylor expanded in x around inf

   \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot {\left(1 + \varepsilon\right)}^{3} + \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right)}, 2\right) \]
10. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot {\left(1 + \varepsilon\right)}^{3} + \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right)}, 2\right) \]

    2. unpow2 N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-1}{6} \cdot {\left(1 + \varepsilon\right)}^{3} + \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right), 2\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-1}{6} \cdot {\left(1 + \varepsilon\right)}^{3} + \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right), 2\right) \]

    4. cube-mult N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(\left(1 + \varepsilon\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)} + \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right), 2\right) \]

    5. unpow2 N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \left(\frac{-1}{6} \cdot \left(\left(1 + \varepsilon\right) \cdot \color{blue}{{\left(1 + \varepsilon\right)}^{2}}\right) + \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right), 2\right) \]

    6. associate-*r* N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot \left(1 + \varepsilon\right)\right) \cdot {\left(1 + \varepsilon\right)}^{2}} + \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right), 2\right) \]

    7. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6} \cdot \left(1 + \varepsilon\right), {\left(1 + \varepsilon\right)}^{2}, \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right)}, 2\right) \]

    8. +-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{-1}{6} \cdot \color{blue}{\left(\varepsilon + 1\right)}, {\left(1 + \varepsilon\right)}^{2}, \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right), 2\right) \]

    9. distribute-rgt-in N/A

       \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{\varepsilon \cdot \frac{-1}{6} + 1 \cdot \frac{-1}{6}}, {\left(1 + \varepsilon\right)}^{2}, \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right), 2\right) \]

    10. metadata-eval N/A

        \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(\varepsilon \cdot \frac{-1}{6} + \color{blue}{\frac{-1}{6}}, {\left(1 + \varepsilon\right)}^{2}, \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right), 2\right) \]

    11. accelerator-lowering-fma.f64 N/A

        \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\varepsilon, \frac{-1}{6}, \frac{-1}{6}\right)}, {\left(1 + \varepsilon\right)}^{2}, \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right), 2\right) \]

    12. unpow2 N/A

        \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \frac{-1}{6}, \frac{-1}{6}\right), \color{blue}{\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)}, \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right), 2\right) \]

    13. *-lowering-*.f64 N/A

        \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \frac{-1}{6}, \frac{-1}{6}\right), \color{blue}{\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)}, \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right), 2\right) \]

    14. +-lowering-+.f64 N/A

        \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \frac{-1}{6}, \frac{-1}{6}\right), \color{blue}{\left(1 + \varepsilon\right)} \cdot \left(1 + \varepsilon\right), \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right), 2\right) \]

    15. +-lowering-+.f64 N/A

        \[\leadsto \frac{1}{2} \cdot \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \frac{-1}{6}, \frac{-1}{6}\right), \left(1 + \varepsilon\right) \cdot \color{blue}{\left(1 + \varepsilon\right)}, \frac{1}{6} \cdot {\left(\varepsilon - 1\right)}^{3}\right), 2\right) \]

    16. cube-mult N/A

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

    17. unpow2 N/A

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

    18. associate-*r* N/A

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

    19. *-lowering-*.f64 N/A

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

11. Simplified 46.1%

    \[\leadsto 0.5 \cdot \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, -0.16666666666666666, -0.16666666666666666\right), \left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right), \mathsf{fma}\left(\varepsilon, 0.16666666666666666, -0.16666666666666666\right) \cdot \left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right)\right)}, 2\right) \]

12. Taylor expanded in eps around 0

    \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(2 + \frac{-1}{3} \cdot {x}^{3}\right)} \]
13. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{-1}{3} \cdot {x}^{3} + 2\right)} \]

    2. distribute-lft-in N/A

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\frac{-1}{3} \cdot {x}^{3}\right) + \frac{1}{2} \cdot 2} \]

    3. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{-1}{3}\right) \cdot {x}^{3}} + \frac{1}{2} \cdot 2 \]

    4. metadata-eval N/A

       \[\leadsto \color{blue}{\frac{-1}{6}} \cdot {x}^{3} + \frac{1}{2} \cdot 2 \]

    5. metadata-eval N/A

       \[\leadsto \frac{-1}{6} \cdot {x}^{3} + \color{blue}{1} \]

    6. accelerator-lowering-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {x}^{3}, 1\right)} \]

    7. cube-mult N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{x \cdot \left(x \cdot x\right)}, 1\right) \]

    8. unpow2 N/A

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

    9. *-lowering-*.f64 N/A

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

    10. unpow2 N/A

        \[\leadsto \mathsf{fma}\left(\frac{-1}{6}, x \cdot \color{blue}{\left(x \cdot x\right)}, 1\right) \]

    11. *-lowering-*.f64 56.6

        \[\leadsto \mathsf{fma}\left(-0.16666666666666666, x \cdot \color{blue}{\left(x \cdot x\right)}, 1\right) \]

14. Simplified 56.6%

    \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, x \cdot \left(x \cdot x\right), 1\right)} \]
15. Add Preprocessing

Alternative 12: 44.1% accurate, 273.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 1.0)

C

double code(double x, double eps) {
	return 1.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 1.0d0
end function

Java

public static double code(double x, double eps) {
	return 1.0;
}

Python

def code(x, eps):
	return 1.0

Julia

function code(x, eps)
	return 1.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 1.0;
end

Wolfram

code[x_, eps_] := 1.0

Derivation

1. Initial program 71.6%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Simplified 47.6%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x)))) (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x))))) 2.0))