expax (section 3.5)

Percentage Accurate: 54.5% → 100.0%

Time: 9.8s

Alternatives: 10

Speedup: 35.0×

Specification

\[710 > a \cdot x\]

Math

\[\begin{array}{l} \\ e^{a \cdot x} - 1 \end{array} \]

FPCore

(FPCore (a x) :precision binary64 (- (exp (* a x)) 1.0))

C

double code(double a, double x) {
	return exp((a * x)) - 1.0;
}

Fortran

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

Java

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

Python

def code(a, x):
	return math.exp((a * x)) - 1.0

Julia

function code(a, x)
	return Float64(exp(Float64(a * x)) - 1.0)
end

MATLAB

function tmp = code(a, x)
	tmp = exp((a * x)) - 1.0;
end

Wolfram

code[a_, x_] := N[(N[Exp[N[(a * x), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 54.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{a \cdot x} - 1 \end{array} \]

FPCore

(FPCore (a x) :precision binary64 (- (exp (* a x)) 1.0))

C

double code(double a, double x) {
	return exp((a * x)) - 1.0;
}

Fortran

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

Java

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

Python

def code(a, x):
	return math.exp((a * x)) - 1.0

Julia

function code(a, x)
	return Float64(exp(Float64(a * x)) - 1.0)
end

MATLAB

function tmp = code(a, x)
	tmp = exp((a * x)) - 1.0;
end

Wolfram

code[a_, x_] := N[(N[Exp[N[(a * x), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double x) {
	return expm1((a * x));
}

Java

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

Python

def code(a, x):
	return math.expm1((a * x))

Julia

function code(a, x)
	return expm1(Float64(a * x))
end

Wolfram

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

Derivation

1. Initial program 57.9%

   \[e^{a \cdot x} - 1 \]
2. Step-by-step derivation

   1. expm1-define N/A

      \[\leadsto \mathsf{expm1}\left(a \cdot x\right) \]

   2. expm1-lowering-expm1.f64 N/A

      \[\leadsto \mathsf{expm1.f64}\left(\left(a \cdot x\right)\right) \]

   3. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, x\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 98.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\\ \mathbf{if}\;a \cdot x \leq -10:\\ \;\;\;\;\frac{1}{1 + a \cdot \left(a \cdot \left(x \cdot x\right) - x\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot x\right) \cdot \left(1 - t\_0 \cdot \left(t\_0 \cdot \left(a \cdot \left(x \cdot \left(a \cdot x\right)\right)\right)\right)\right)}{1 - a \cdot \left(x \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (let* ((t_0 (+ 0.5 (* a (* x 0.16666666666666666)))))
   (if (<= (* a x) -10.0)
     (+ (/ 1.0 (+ 1.0 (* a (- (* a (* x x)) x)))) -1.0)
     (/
      (* (* a x) (- 1.0 (* t_0 (* t_0 (* a (* x (* a x)))))))
      (- 1.0 (* a (* x t_0)))))))

C

double code(double a, double x) {
	double t_0 = 0.5 + (a * (x * 0.16666666666666666));
	double tmp;
	if ((a * x) <= -10.0) {
		tmp = (1.0 / (1.0 + (a * ((a * (x * x)) - x)))) + -1.0;
	} else {
		tmp = ((a * x) * (1.0 - (t_0 * (t_0 * (a * (x * (a * x))))))) / (1.0 - (a * (x * t_0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 + (a * (x * 0.16666666666666666d0))
    if ((a * x) <= (-10.0d0)) then
        tmp = (1.0d0 / (1.0d0 + (a * ((a * (x * x)) - x)))) + (-1.0d0)
    else
        tmp = ((a * x) * (1.0d0 - (t_0 * (t_0 * (a * (x * (a * x))))))) / (1.0d0 - (a * (x * t_0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double t_0 = 0.5 + (a * (x * 0.16666666666666666));
	double tmp;
	if ((a * x) <= -10.0) {
		tmp = (1.0 / (1.0 + (a * ((a * (x * x)) - x)))) + -1.0;
	} else {
		tmp = ((a * x) * (1.0 - (t_0 * (t_0 * (a * (x * (a * x))))))) / (1.0 - (a * (x * t_0)));
	}
	return tmp;
}

Python

def code(a, x):
	t_0 = 0.5 + (a * (x * 0.16666666666666666))
	tmp = 0
	if (a * x) <= -10.0:
		tmp = (1.0 / (1.0 + (a * ((a * (x * x)) - x)))) + -1.0
	else:
		tmp = ((a * x) * (1.0 - (t_0 * (t_0 * (a * (x * (a * x))))))) / (1.0 - (a * (x * t_0)))
	return tmp

Julia

function code(a, x)
	t_0 = Float64(0.5 + Float64(a * Float64(x * 0.16666666666666666)))
	tmp = 0.0
	if (Float64(a * x) <= -10.0)
		tmp = Float64(Float64(1.0 / Float64(1.0 + Float64(a * Float64(Float64(a * Float64(x * x)) - x)))) + -1.0);
	else
		tmp = Float64(Float64(Float64(a * x) * Float64(1.0 - Float64(t_0 * Float64(t_0 * Float64(a * Float64(x * Float64(a * x))))))) / Float64(1.0 - Float64(a * Float64(x * t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	t_0 = 0.5 + (a * (x * 0.16666666666666666));
	tmp = 0.0;
	if ((a * x) <= -10.0)
		tmp = (1.0 / (1.0 + (a * ((a * (x * x)) - x)))) + -1.0;
	else
		tmp = ((a * x) * (1.0 - (t_0 * (t_0 * (a * (x * (a * x))))))) / (1.0 - (a * (x * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := Block[{t$95$0 = N[(0.5 + N[(a * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a * x), $MachinePrecision], -10.0], N[(N[(1.0 / N[(1.0 + N[(a * N[(N[(a * N[(x * x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(a * x), $MachinePrecision] * N[(1.0 - N[(t$95$0 * N[(t$95$0 * N[(a * N[(x * N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(a * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -10

   1. Initial program 100.0%

      \[e^{a \cdot x} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + a \cdot x\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(a \cdot x + 1\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(a \cdot x\right), 1\right), 1\right) \]

      3. *-lowering-*.f64 5.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), 1\right), 1\right) \]

   5. Simplified 5.2%

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

      1. flip-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1}{a \cdot x - 1}\right), 1\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\frac{a \cdot x - 1}{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1}}\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{a \cdot x - 1}{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1}\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(a \cdot x - 1\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(a \cdot x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a \cdot x\right), \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1\right)\right)\right), 1\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\left(\left(x \cdot a\right) \cdot \left(a \cdot x\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\left(x \cdot \left(a \cdot \left(a \cdot x\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(a \cdot \left(a \cdot x\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \left(a \cdot x\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

      17. metadata-eval 4.2%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, x\right)\right)\right), -1\right)\right)\right), 1\right) \]

   7. Applied egg-rr 4.2%

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

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(1 + a \cdot \left(-1 \cdot x + a \cdot {x}^{2}\right)\right)}\right), 1\right) \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \left(a \cdot \left(-1 \cdot x\right) + a \cdot \left(a \cdot {x}^{2}\right)\right)\right)\right), 1\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \left(\left(a \cdot -1\right) \cdot x + a \cdot \left(a \cdot {x}^{2}\right)\right)\right)\right), 1\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \left(\left(-1 \cdot a\right) \cdot x + a \cdot \left(a \cdot {x}^{2}\right)\right)\right)\right), 1\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\left(1 + \left(-1 \cdot a\right) \cdot x\right) + a \cdot \left(a \cdot {x}^{2}\right)\right)\right), 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\left(1 + \left(-1 \cdot a\right) \cdot x\right) + \left(a \cdot a\right) \cdot {x}^{2}\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\left(1 + \left(-1 \cdot a\right) \cdot x\right) + {a}^{2} \cdot {x}^{2}\right)\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\left(1 + \left(-1 \cdot a\right) \cdot x\right) + {a}^{2} \cdot \left(x \cdot x\right)\right)\right), 1\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\left(1 + \left(-1 \cdot a\right) \cdot x\right) + \left({a}^{2} \cdot x\right) \cdot x\right)\right), 1\right) \]

      9. associate-+r+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \left(\left(-1 \cdot a\right) \cdot x + \left({a}^{2} \cdot x\right) \cdot x\right)\right)\right), 1\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(1 + x \cdot \left(-1 \cdot a + {a}^{2} \cdot x\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(x \cdot \left(-1 \cdot a + {a}^{2} \cdot x\right)\right)\right)\right), 1\right) \]

      12. distribute-rgt-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\left(-1 \cdot a\right) \cdot x + \left({a}^{2} \cdot x\right) \cdot x\right)\right)\right), 1\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\left(-1 \cdot a\right) \cdot x + {a}^{2} \cdot \left(x \cdot x\right)\right)\right)\right), 1\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\left(-1 \cdot a\right) \cdot x + {a}^{2} \cdot {x}^{2}\right)\right)\right), 1\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\left(a \cdot -1\right) \cdot x + {a}^{2} \cdot {x}^{2}\right)\right)\right), 1\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(a \cdot \left(-1 \cdot x\right) + {a}^{2} \cdot {x}^{2}\right)\right)\right), 1\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(a \cdot \left(-1 \cdot x\right) + \left(a \cdot a\right) \cdot {x}^{2}\right)\right)\right), 1\right) \]

      18. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(a \cdot \left(-1 \cdot x\right) + a \cdot \left(a \cdot {x}^{2}\right)\right)\right)\right), 1\right) \]

      19. distribute-lft-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(a \cdot \left(-1 \cdot x + a \cdot {x}^{2}\right)\right)\right)\right), 1\right) \]

   10. Simplified 98.9%

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

   if -10 < (*.f64 a x)

   1. Initial program 36.7%

      \[e^{a \cdot x} - 1 \]
   2. Step-by-step derivation

      1. expm1-define N/A

         \[\leadsto \mathsf{expm1}\left(a \cdot x\right) \]

      2. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{expm1.f64}\left(\left(a \cdot x\right)\right) \]

      3. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

         \[\leadsto a \cdot \left(\left(a \cdot \left(\frac{1}{6} \cdot \left(a \cdot {x}^{3}\right)\right) + a \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right) + x\right) \]

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. unpow3 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

         \[\leadsto a \cdot \left(\left(\left(a \cdot \left(\frac{1}{6} \cdot a\right)\right) \cdot {x}^{2}\right) \cdot x + \left(\left(a \cdot \frac{1}{2}\right) \cdot {x}^{2} + x\right)\right) \]

      10. *-commutative N/A

          \[\leadsto a \cdot \left(\left(\left(a \cdot \left(\frac{1}{6} \cdot a\right)\right) \cdot {x}^{2}\right) \cdot x + \left(\left(\frac{1}{2} \cdot a\right) \cdot {x}^{2} + x\right)\right) \]

      11. unpow2 N/A

          \[\leadsto a \cdot \left(\left(\left(a \cdot \left(\frac{1}{6} \cdot a\right)\right) \cdot {x}^{2}\right) \cdot x + \left(\left(\frac{1}{2} \cdot a\right) \cdot \left(x \cdot x\right) + x\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto a \cdot \left(\left(\left(a \cdot \left(\frac{1}{6} \cdot a\right)\right) \cdot {x}^{2}\right) \cdot x + \left(\left(\left(\frac{1}{2} \cdot a\right) \cdot x\right) \cdot x + x\right)\right) \]

      13. distribute-lft1-in N/A

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

      14. distribute-rgt-out N/A

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

   7. Simplified 99.3%

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

      1. *-commutative N/A

         \[\leadsto \left(1 + \left(a \cdot x\right) \cdot \left(\frac{1}{2} + \left(a \cdot x\right) \cdot \frac{1}{6}\right)\right) \cdot \color{blue}{\left(a \cdot x\right)} \]

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 \cdot 1 - \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + \left(a \cdot x\right) \cdot \frac{1}{6}\right)\right) \cdot \left(\left(a \cdot x\right) \cdot \left(\frac{1}{2} + \left(a \cdot x\right) \cdot \frac{1}{6}\right)\right)\right) \cdot \left(a \cdot x\right)\right), \color{blue}{\left(1 - \left(a \cdot x\right) \cdot \left(\frac{1}{2} + \left(a \cdot x\right) \cdot \frac{1}{6}\right)\right)}\right) \]

   9. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\frac{\left(1 - \left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right) \cdot \left(\left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right) \cdot \left(a \cdot \left(x \cdot \left(a \cdot x\right)\right)\right)\right)\right) \cdot \left(a \cdot x\right)}{1 - a \cdot \left(x \cdot \left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -10:\\ \;\;\;\;\frac{1}{1 + a \cdot \left(a \cdot \left(x \cdot x\right) - x\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot x\right) \cdot \left(1 - \left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right) \cdot \left(\left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right) \cdot \left(a \cdot \left(x \cdot \left(a \cdot x\right)\right)\right)\right)\right)}{1 - a \cdot \left(x \cdot \left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -10:\\ \;\;\;\;\frac{1}{1 + a \cdot \left(a \cdot \left(x \cdot x\right) - x\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(a \cdot x\right) \cdot \left(0.5 + \left(a \cdot x\right) \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -10.0)
   (+ (/ 1.0 (+ 1.0 (* a (- (* a (* x x)) x)))) -1.0)
   (* (* a x) (+ 1.0 (* (* a x) (+ 0.5 (* (* a x) 0.16666666666666666)))))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -10.0) {
		tmp = (1.0 / (1.0 + (a * ((a * (x * x)) - x)))) + -1.0;
	} else {
		tmp = (a * x) * (1.0 + ((a * x) * (0.5 + ((a * x) * 0.16666666666666666))));
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((a * x) <= (-10.0d0)) then
        tmp = (1.0d0 / (1.0d0 + (a * ((a * (x * x)) - x)))) + (-1.0d0)
    else
        tmp = (a * x) * (1.0d0 + ((a * x) * (0.5d0 + ((a * x) * 0.16666666666666666d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -10.0) {
		tmp = (1.0 / (1.0 + (a * ((a * (x * x)) - x)))) + -1.0;
	} else {
		tmp = (a * x) * (1.0 + ((a * x) * (0.5 + ((a * x) * 0.16666666666666666))));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -10.0:
		tmp = (1.0 / (1.0 + (a * ((a * (x * x)) - x)))) + -1.0
	else:
		tmp = (a * x) * (1.0 + ((a * x) * (0.5 + ((a * x) * 0.16666666666666666))))
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -10.0)
		tmp = Float64(Float64(1.0 / Float64(1.0 + Float64(a * Float64(Float64(a * Float64(x * x)) - x)))) + -1.0);
	else
		tmp = Float64(Float64(a * x) * Float64(1.0 + Float64(Float64(a * x) * Float64(0.5 + Float64(Float64(a * x) * 0.16666666666666666)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -10.0)
		tmp = (1.0 / (1.0 + (a * ((a * (x * x)) - x)))) + -1.0;
	else
		tmp = (a * x) * (1.0 + ((a * x) * (0.5 + ((a * x) * 0.16666666666666666))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -10.0], N[(N[(1.0 / N[(1.0 + N[(a * N[(N[(a * N[(x * x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(a * x), $MachinePrecision] * N[(1.0 + N[(N[(a * x), $MachinePrecision] * N[(0.5 + N[(N[(a * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -10

   1. Initial program 100.0%

      \[e^{a \cdot x} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + a \cdot x\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(a \cdot x + 1\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(a \cdot x\right), 1\right), 1\right) \]

      3. *-lowering-*.f64 5.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), 1\right), 1\right) \]

   5. Simplified 5.2%

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

      1. flip-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1}{a \cdot x - 1}\right), 1\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\frac{a \cdot x - 1}{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1}}\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{a \cdot x - 1}{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1}\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(a \cdot x - 1\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(a \cdot x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a \cdot x\right), \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1\right)\right)\right), 1\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\left(\left(x \cdot a\right) \cdot \left(a \cdot x\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\left(x \cdot \left(a \cdot \left(a \cdot x\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(a \cdot \left(a \cdot x\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \left(a \cdot x\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

      17. metadata-eval 4.2%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, x\right)\right)\right), -1\right)\right)\right), 1\right) \]

   7. Applied egg-rr 4.2%

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

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(1 + a \cdot \left(-1 \cdot x + a \cdot {x}^{2}\right)\right)}\right), 1\right) \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \left(a \cdot \left(-1 \cdot x\right) + a \cdot \left(a \cdot {x}^{2}\right)\right)\right)\right), 1\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \left(\left(a \cdot -1\right) \cdot x + a \cdot \left(a \cdot {x}^{2}\right)\right)\right)\right), 1\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \left(\left(-1 \cdot a\right) \cdot x + a \cdot \left(a \cdot {x}^{2}\right)\right)\right)\right), 1\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\left(1 + \left(-1 \cdot a\right) \cdot x\right) + a \cdot \left(a \cdot {x}^{2}\right)\right)\right), 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\left(1 + \left(-1 \cdot a\right) \cdot x\right) + \left(a \cdot a\right) \cdot {x}^{2}\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\left(1 + \left(-1 \cdot a\right) \cdot x\right) + {a}^{2} \cdot {x}^{2}\right)\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\left(1 + \left(-1 \cdot a\right) \cdot x\right) + {a}^{2} \cdot \left(x \cdot x\right)\right)\right), 1\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\left(1 + \left(-1 \cdot a\right) \cdot x\right) + \left({a}^{2} \cdot x\right) \cdot x\right)\right), 1\right) \]

      9. associate-+r+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \left(\left(-1 \cdot a\right) \cdot x + \left({a}^{2} \cdot x\right) \cdot x\right)\right)\right), 1\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(1 + x \cdot \left(-1 \cdot a + {a}^{2} \cdot x\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(x \cdot \left(-1 \cdot a + {a}^{2} \cdot x\right)\right)\right)\right), 1\right) \]

      12. distribute-rgt-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\left(-1 \cdot a\right) \cdot x + \left({a}^{2} \cdot x\right) \cdot x\right)\right)\right), 1\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\left(-1 \cdot a\right) \cdot x + {a}^{2} \cdot \left(x \cdot x\right)\right)\right)\right), 1\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\left(-1 \cdot a\right) \cdot x + {a}^{2} \cdot {x}^{2}\right)\right)\right), 1\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\left(a \cdot -1\right) \cdot x + {a}^{2} \cdot {x}^{2}\right)\right)\right), 1\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(a \cdot \left(-1 \cdot x\right) + {a}^{2} \cdot {x}^{2}\right)\right)\right), 1\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(a \cdot \left(-1 \cdot x\right) + \left(a \cdot a\right) \cdot {x}^{2}\right)\right)\right), 1\right) \]

      18. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(a \cdot \left(-1 \cdot x\right) + a \cdot \left(a \cdot {x}^{2}\right)\right)\right)\right), 1\right) \]

      19. distribute-lft-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(a \cdot \left(-1 \cdot x + a \cdot {x}^{2}\right)\right)\right)\right), 1\right) \]

   10. Simplified 98.9%

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

   if -10 < (*.f64 a x)

   1. Initial program 36.7%

      \[e^{a \cdot x} - 1 \]
   2. Step-by-step derivation

      1. expm1-define N/A

         \[\leadsto \mathsf{expm1}\left(a \cdot x\right) \]

      2. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{expm1.f64}\left(\left(a \cdot x\right)\right) \]

      3. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

         \[\leadsto a \cdot \left(\left(a \cdot \left(\frac{1}{6} \cdot \left(a \cdot {x}^{3}\right)\right) + a \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)\right) + x\right) \]

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. unpow3 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

         \[\leadsto a \cdot \left(\left(\left(a \cdot \left(\frac{1}{6} \cdot a\right)\right) \cdot {x}^{2}\right) \cdot x + \left(\left(a \cdot \frac{1}{2}\right) \cdot {x}^{2} + x\right)\right) \]

      10. *-commutative N/A

          \[\leadsto a \cdot \left(\left(\left(a \cdot \left(\frac{1}{6} \cdot a\right)\right) \cdot {x}^{2}\right) \cdot x + \left(\left(\frac{1}{2} \cdot a\right) \cdot {x}^{2} + x\right)\right) \]

      11. unpow2 N/A

          \[\leadsto a \cdot \left(\left(\left(a \cdot \left(\frac{1}{6} \cdot a\right)\right) \cdot {x}^{2}\right) \cdot x + \left(\left(\frac{1}{2} \cdot a\right) \cdot \left(x \cdot x\right) + x\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto a \cdot \left(\left(\left(a \cdot \left(\frac{1}{6} \cdot a\right)\right) \cdot {x}^{2}\right) \cdot x + \left(\left(\left(\frac{1}{2} \cdot a\right) \cdot x\right) \cdot x + x\right)\right) \]

      13. distribute-lft1-in N/A

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

      14. distribute-rgt-out N/A

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

   7. Simplified 99.3%

      \[\leadsto \color{blue}{\left(a \cdot x\right) \cdot \left(1 + \left(a \cdot x\right) \cdot \left(0.5 + \left(a \cdot x\right) \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -10:\\ \;\;\;\;\frac{1}{1 + a \cdot \left(a \cdot \left(x \cdot x\right) - x\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(a \cdot x\right) \cdot \left(0.5 + \left(a \cdot x\right) \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -10:\\ \;\;\;\;\frac{1}{1 + a \cdot \left(a \cdot \left(x \cdot x\right) - x\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(1 + a \cdot \left(x \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -10.0)
   (+ (/ 1.0 (+ 1.0 (* a (- (* a (* x x)) x)))) -1.0)
   (* x (* a (+ 1.0 (* a (* x 0.5)))))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -10.0) {
		tmp = (1.0 / (1.0 + (a * ((a * (x * x)) - x)))) + -1.0;
	} else {
		tmp = x * (a * (1.0 + (a * (x * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((a * x) <= (-10.0d0)) then
        tmp = (1.0d0 / (1.0d0 + (a * ((a * (x * x)) - x)))) + (-1.0d0)
    else
        tmp = x * (a * (1.0d0 + (a * (x * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -10.0) {
		tmp = (1.0 / (1.0 + (a * ((a * (x * x)) - x)))) + -1.0;
	} else {
		tmp = x * (a * (1.0 + (a * (x * 0.5))));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -10.0:
		tmp = (1.0 / (1.0 + (a * ((a * (x * x)) - x)))) + -1.0
	else:
		tmp = x * (a * (1.0 + (a * (x * 0.5))))
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -10.0)
		tmp = Float64(Float64(1.0 / Float64(1.0 + Float64(a * Float64(Float64(a * Float64(x * x)) - x)))) + -1.0);
	else
		tmp = Float64(x * Float64(a * Float64(1.0 + Float64(a * Float64(x * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -10.0)
		tmp = (1.0 / (1.0 + (a * ((a * (x * x)) - x)))) + -1.0;
	else
		tmp = x * (a * (1.0 + (a * (x * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -10.0], N[(N[(1.0 / N[(1.0 + N[(a * N[(N[(a * N[(x * x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(x * N[(a * N[(1.0 + N[(a * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -10

   1. Initial program 100.0%

      \[e^{a \cdot x} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + a \cdot x\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(a \cdot x + 1\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(a \cdot x\right), 1\right), 1\right) \]

      3. *-lowering-*.f64 5.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), 1\right), 1\right) \]

   5. Simplified 5.2%

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

      1. flip-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1}{a \cdot x - 1}\right), 1\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\frac{a \cdot x - 1}{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1}}\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{a \cdot x - 1}{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1}\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(a \cdot x - 1\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(a \cdot x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a \cdot x\right), \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1\right)\right)\right), 1\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\left(\left(x \cdot a\right) \cdot \left(a \cdot x\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\left(x \cdot \left(a \cdot \left(a \cdot x\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(a \cdot \left(a \cdot x\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \left(a \cdot x\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

      17. metadata-eval 4.2%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, x\right)\right)\right), -1\right)\right)\right), 1\right) \]

   7. Applied egg-rr 4.2%

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

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(1 + a \cdot \left(-1 \cdot x + a \cdot {x}^{2}\right)\right)}\right), 1\right) \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \left(a \cdot \left(-1 \cdot x\right) + a \cdot \left(a \cdot {x}^{2}\right)\right)\right)\right), 1\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \left(\left(a \cdot -1\right) \cdot x + a \cdot \left(a \cdot {x}^{2}\right)\right)\right)\right), 1\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \left(\left(-1 \cdot a\right) \cdot x + a \cdot \left(a \cdot {x}^{2}\right)\right)\right)\right), 1\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\left(1 + \left(-1 \cdot a\right) \cdot x\right) + a \cdot \left(a \cdot {x}^{2}\right)\right)\right), 1\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\left(1 + \left(-1 \cdot a\right) \cdot x\right) + \left(a \cdot a\right) \cdot {x}^{2}\right)\right), 1\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\left(1 + \left(-1 \cdot a\right) \cdot x\right) + {a}^{2} \cdot {x}^{2}\right)\right), 1\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\left(1 + \left(-1 \cdot a\right) \cdot x\right) + {a}^{2} \cdot \left(x \cdot x\right)\right)\right), 1\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\left(1 + \left(-1 \cdot a\right) \cdot x\right) + \left({a}^{2} \cdot x\right) \cdot x\right)\right), 1\right) \]

      9. associate-+r+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \left(\left(-1 \cdot a\right) \cdot x + \left({a}^{2} \cdot x\right) \cdot x\right)\right)\right), 1\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(1 + x \cdot \left(-1 \cdot a + {a}^{2} \cdot x\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(x \cdot \left(-1 \cdot a + {a}^{2} \cdot x\right)\right)\right)\right), 1\right) \]

      12. distribute-rgt-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\left(-1 \cdot a\right) \cdot x + \left({a}^{2} \cdot x\right) \cdot x\right)\right)\right), 1\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\left(-1 \cdot a\right) \cdot x + {a}^{2} \cdot \left(x \cdot x\right)\right)\right)\right), 1\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\left(-1 \cdot a\right) \cdot x + {a}^{2} \cdot {x}^{2}\right)\right)\right), 1\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\left(a \cdot -1\right) \cdot x + {a}^{2} \cdot {x}^{2}\right)\right)\right), 1\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(a \cdot \left(-1 \cdot x\right) + {a}^{2} \cdot {x}^{2}\right)\right)\right), 1\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(a \cdot \left(-1 \cdot x\right) + \left(a \cdot a\right) \cdot {x}^{2}\right)\right)\right), 1\right) \]

      18. associate-*r* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(a \cdot \left(-1 \cdot x\right) + a \cdot \left(a \cdot {x}^{2}\right)\right)\right)\right), 1\right) \]

      19. distribute-lft-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(a \cdot \left(-1 \cdot x + a \cdot {x}^{2}\right)\right)\right)\right), 1\right) \]

   10. Simplified 98.9%

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

   if -10 < (*.f64 a x)

   1. Initial program 36.7%

      \[e^{a \cdot x} - 1 \]
   2. Step-by-step derivation

      1. expm1-define N/A

         \[\leadsto \mathsf{expm1}\left(a \cdot x\right) \]

      2. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{expm1.f64}\left(\left(a \cdot x\right)\right) \]

      3. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left(x + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot a\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(x + \frac{1}{2} \cdot \color{blue}{\left({x}^{2} \cdot a\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(x + \frac{1}{2} \cdot \left(a \cdot \color{blue}{{x}^{2}}\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(x + \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{{x}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(x + \left(\frac{1}{2} \cdot a\right) \cdot \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(x + \left(\left(\frac{1}{2} \cdot a\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \]

      9. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(\left(\left(\frac{1}{2} \cdot a\right) \cdot x + 1\right) \cdot \color{blue}{x}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(a, \left(x \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot x + 1\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot x + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \left(1 + \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot x}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot x\right)}\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot x\right)}\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(a \cdot x\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(a \cdot x\right), \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 98.8%

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \frac{1}{2}\right)\right)\right)\right) \]

   7. Simplified 98.8%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(a \cdot \left(1 + \left(a \cdot x\right) \cdot \frac{1}{2}\right)\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(1 + \left(a \cdot x\right) \cdot \frac{1}{2}\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(a \cdot x\right) \cdot \frac{1}{2}\right)\right)\right), x\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \left(a \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(x \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

      8. *-lowering-*.f64 98.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right), x\right) \]

   9. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\left(a \cdot \left(1 + a \cdot \left(x \cdot 0.5\right)\right)\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -10:\\ \;\;\;\;\frac{1}{1 + a \cdot \left(a \cdot \left(x \cdot x\right) - x\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(1 + a \cdot \left(x \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -10:\\ \;\;\;\;\frac{1}{1 - a \cdot x} + -1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(1 + a \cdot \left(x \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -10.0)
   (+ (/ 1.0 (- 1.0 (* a x))) -1.0)
   (* x (* a (+ 1.0 (* a (* x 0.5)))))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -10.0) {
		tmp = (1.0 / (1.0 - (a * x))) + -1.0;
	} else {
		tmp = x * (a * (1.0 + (a * (x * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((a * x) <= (-10.0d0)) then
        tmp = (1.0d0 / (1.0d0 - (a * x))) + (-1.0d0)
    else
        tmp = x * (a * (1.0d0 + (a * (x * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -10.0) {
		tmp = (1.0 / (1.0 - (a * x))) + -1.0;
	} else {
		tmp = x * (a * (1.0 + (a * (x * 0.5))));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -10.0:
		tmp = (1.0 / (1.0 - (a * x))) + -1.0
	else:
		tmp = x * (a * (1.0 + (a * (x * 0.5))))
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -10.0)
		tmp = Float64(Float64(1.0 / Float64(1.0 - Float64(a * x))) + -1.0);
	else
		tmp = Float64(x * Float64(a * Float64(1.0 + Float64(a * Float64(x * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -10.0)
		tmp = (1.0 / (1.0 - (a * x))) + -1.0;
	else
		tmp = x * (a * (1.0 + (a * (x * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -10.0], N[(N[(1.0 / N[(1.0 - N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(x * N[(a * N[(1.0 + N[(a * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -10

   1. Initial program 100.0%

      \[e^{a \cdot x} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + a \cdot x\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(a \cdot x + 1\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(a \cdot x\right), 1\right), 1\right) \]

      3. *-lowering-*.f64 5.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), 1\right), 1\right) \]

   5. Simplified 5.2%

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

      1. flip-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1}{a \cdot x - 1}\right), 1\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\frac{a \cdot x - 1}{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1}}\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{a \cdot x - 1}{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1}\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(a \cdot x - 1\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(a \cdot x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a \cdot x\right), \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1\right)\right)\right), 1\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\left(\left(x \cdot a\right) \cdot \left(a \cdot x\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\left(x \cdot \left(a \cdot \left(a \cdot x\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(a \cdot \left(a \cdot x\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \left(a \cdot x\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

      17. metadata-eval 4.2%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, x\right)\right)\right), -1\right)\right)\right), 1\right) \]

   7. Applied egg-rr 4.2%

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

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(1 + -1 \cdot \left(a \cdot x\right)\right)}\right), 1\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \left(\mathsf{neg}\left(a \cdot x\right)\right)\right)\right), 1\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(1 - a \cdot x\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \left(a \cdot x\right)\right)\right), 1\right) \]

      4. *-lowering-*.f64 97.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), 1\right) \]

   10. Simplified 97.6%

       \[\leadsto \frac{1}{\color{blue}{1 - a \cdot x}} - 1 \]

   if -10 < (*.f64 a x)

   1. Initial program 36.7%

      \[e^{a \cdot x} - 1 \]
   2. Step-by-step derivation

      1. expm1-define N/A

         \[\leadsto \mathsf{expm1}\left(a \cdot x\right) \]

      2. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{expm1.f64}\left(\left(a \cdot x\right)\right) \]

      3. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left(x + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot a\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(x + \frac{1}{2} \cdot \color{blue}{\left({x}^{2} \cdot a\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(x + \frac{1}{2} \cdot \left(a \cdot \color{blue}{{x}^{2}}\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(x + \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{{x}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(x + \left(\frac{1}{2} \cdot a\right) \cdot \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(x + \left(\left(\frac{1}{2} \cdot a\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \]

      9. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(\left(\left(\frac{1}{2} \cdot a\right) \cdot x + 1\right) \cdot \color{blue}{x}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(a, \left(x \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot x + 1\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot x + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \left(1 + \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot x}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot x\right)}\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot x\right)}\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(a \cdot x\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(a \cdot x\right), \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 98.8%

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \frac{1}{2}\right)\right)\right)\right) \]

   7. Simplified 98.8%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(a \cdot \left(1 + \left(a \cdot x\right) \cdot \frac{1}{2}\right)\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \left(1 + \left(a \cdot x\right) \cdot \frac{1}{2}\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \left(\left(a \cdot x\right) \cdot \frac{1}{2}\right)\right)\right), x\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \left(a \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \left(x \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

      8. *-lowering-*.f64 98.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right), x\right) \]

   9. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\left(a \cdot \left(1 + a \cdot \left(x \cdot 0.5\right)\right)\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -10:\\ \;\;\;\;\frac{1}{1 - a \cdot x} + -1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a \cdot \left(1 + a \cdot \left(x \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.3% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -10:\\ \;\;\;\;\frac{1}{1 - a \cdot x} + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x + x \cdot \left(a \cdot \left(x \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -10.0)
   (+ (/ 1.0 (- 1.0 (* a x))) -1.0)
   (* a (+ x (* x (* a (* x 0.5)))))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -10.0) {
		tmp = (1.0 / (1.0 - (a * x))) + -1.0;
	} else {
		tmp = a * (x + (x * (a * (x * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((a * x) <= (-10.0d0)) then
        tmp = (1.0d0 / (1.0d0 - (a * x))) + (-1.0d0)
    else
        tmp = a * (x + (x * (a * (x * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -10.0) {
		tmp = (1.0 / (1.0 - (a * x))) + -1.0;
	} else {
		tmp = a * (x + (x * (a * (x * 0.5))));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -10.0:
		tmp = (1.0 / (1.0 - (a * x))) + -1.0
	else:
		tmp = a * (x + (x * (a * (x * 0.5))))
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -10.0)
		tmp = Float64(Float64(1.0 / Float64(1.0 - Float64(a * x))) + -1.0);
	else
		tmp = Float64(a * Float64(x + Float64(x * Float64(a * Float64(x * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -10.0)
		tmp = (1.0 / (1.0 - (a * x))) + -1.0;
	else
		tmp = a * (x + (x * (a * (x * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -10.0], N[(N[(1.0 / N[(1.0 - N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(a * N[(x + N[(x * N[(a * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -10

   1. Initial program 100.0%

      \[e^{a \cdot x} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + a \cdot x\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(a \cdot x + 1\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(a \cdot x\right), 1\right), 1\right) \]

      3. *-lowering-*.f64 5.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), 1\right), 1\right) \]

   5. Simplified 5.2%

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

      1. flip-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1}{a \cdot x - 1}\right), 1\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\frac{a \cdot x - 1}{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1}}\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{a \cdot x - 1}{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1}\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(a \cdot x - 1\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(a \cdot x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a \cdot x\right), \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1\right)\right)\right), 1\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\left(\left(x \cdot a\right) \cdot \left(a \cdot x\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\left(x \cdot \left(a \cdot \left(a \cdot x\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(a \cdot \left(a \cdot x\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \left(a \cdot x\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

      17. metadata-eval 4.2%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, x\right)\right)\right), -1\right)\right)\right), 1\right) \]

   7. Applied egg-rr 4.2%

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

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(1 + -1 \cdot \left(a \cdot x\right)\right)}\right), 1\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \left(\mathsf{neg}\left(a \cdot x\right)\right)\right)\right), 1\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(1 - a \cdot x\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \left(a \cdot x\right)\right)\right), 1\right) \]

      4. *-lowering-*.f64 97.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), 1\right) \]

   10. Simplified 97.6%

       \[\leadsto \frac{1}{\color{blue}{1 - a \cdot x}} - 1 \]

   if -10 < (*.f64 a x)

   1. Initial program 36.7%

      \[e^{a \cdot x} - 1 \]
   2. Step-by-step derivation

      1. expm1-define N/A

         \[\leadsto \mathsf{expm1}\left(a \cdot x\right) \]

      2. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{expm1.f64}\left(\left(a \cdot x\right)\right) \]

      3. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left(x + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot a\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(x + \frac{1}{2} \cdot \color{blue}{\left({x}^{2} \cdot a\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(x + \frac{1}{2} \cdot \left(a \cdot \color{blue}{{x}^{2}}\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(x + \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{{x}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(x + \left(\frac{1}{2} \cdot a\right) \cdot \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(x + \left(\left(\frac{1}{2} \cdot a\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \]

      9. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(\left(\left(\frac{1}{2} \cdot a\right) \cdot x + 1\right) \cdot \color{blue}{x}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(a, \left(x \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot x + 1\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot x + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \left(1 + \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot x}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot x\right)}\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot x\right)}\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(a \cdot x\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(a \cdot x\right), \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 98.8%

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \frac{1}{2}\right)\right)\right)\right) \]

   7. Simplified 98.8%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(x \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{2} + \color{blue}{1}\right)\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(\left(\left(a \cdot x\right) \cdot \frac{1}{2}\right) \cdot x + \color{blue}{1 \cdot x}\right)\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(\left(\left(a \cdot x\right) \cdot \frac{1}{2}\right) \cdot x + x\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\left(\left(a \cdot x\right) \cdot \frac{1}{2}\right) \cdot x\right), \color{blue}{x}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(x \cdot \left(\left(a \cdot x\right) \cdot \frac{1}{2}\right)\right), x\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\left(a \cdot x\right) \cdot \frac{1}{2}\right)\right), x\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(a \cdot \left(x \cdot \frac{1}{2}\right)\right)\right), x\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \left(x \cdot \frac{1}{2}\right)\right)\right), x\right)\right) \]

      9. *-lowering-*.f64 98.8%

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right), x\right)\right) \]

   9. Applied egg-rr 98.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -10:\\ \;\;\;\;\frac{1}{1 - a \cdot x} + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x + x \cdot \left(a \cdot \left(x \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.3% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -10:\\ \;\;\;\;\frac{1}{1 - a \cdot x} + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(1 + \left(a \cdot x\right) \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -10.0)
   (+ (/ 1.0 (- 1.0 (* a x))) -1.0)
   (* a (* x (+ 1.0 (* (* a x) 0.5))))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -10.0) {
		tmp = (1.0 / (1.0 - (a * x))) + -1.0;
	} else {
		tmp = a * (x * (1.0 + ((a * x) * 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((a * x) <= (-10.0d0)) then
        tmp = (1.0d0 / (1.0d0 - (a * x))) + (-1.0d0)
    else
        tmp = a * (x * (1.0d0 + ((a * x) * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -10.0) {
		tmp = (1.0 / (1.0 - (a * x))) + -1.0;
	} else {
		tmp = a * (x * (1.0 + ((a * x) * 0.5)));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -10.0:
		tmp = (1.0 / (1.0 - (a * x))) + -1.0
	else:
		tmp = a * (x * (1.0 + ((a * x) * 0.5)))
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -10.0)
		tmp = Float64(Float64(1.0 / Float64(1.0 - Float64(a * x))) + -1.0);
	else
		tmp = Float64(a * Float64(x * Float64(1.0 + Float64(Float64(a * x) * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -10.0)
		tmp = (1.0 / (1.0 - (a * x))) + -1.0;
	else
		tmp = a * (x * (1.0 + ((a * x) * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -10.0], N[(N[(1.0 / N[(1.0 - N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(a * N[(x * N[(1.0 + N[(N[(a * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -10

   1. Initial program 100.0%

      \[e^{a \cdot x} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + a \cdot x\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(a \cdot x + 1\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(a \cdot x\right), 1\right), 1\right) \]

      3. *-lowering-*.f64 5.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), 1\right), 1\right) \]

   5. Simplified 5.2%

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

      1. flip-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1}{a \cdot x - 1}\right), 1\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\frac{a \cdot x - 1}{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1}}\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{a \cdot x - 1}{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1}\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(a \cdot x - 1\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(a \cdot x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a \cdot x\right), \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1\right)\right)\right), 1\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\left(\left(x \cdot a\right) \cdot \left(a \cdot x\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\left(x \cdot \left(a \cdot \left(a \cdot x\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(a \cdot \left(a \cdot x\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \left(a \cdot x\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

      17. metadata-eval 4.2%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, x\right)\right)\right), -1\right)\right)\right), 1\right) \]

   7. Applied egg-rr 4.2%

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

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(1 + -1 \cdot \left(a \cdot x\right)\right)}\right), 1\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \left(\mathsf{neg}\left(a \cdot x\right)\right)\right)\right), 1\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(1 - a \cdot x\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \left(a \cdot x\right)\right)\right), 1\right) \]

      4. *-lowering-*.f64 97.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), 1\right) \]

   10. Simplified 97.6%

       \[\leadsto \frac{1}{\color{blue}{1 - a \cdot x}} - 1 \]

   if -10 < (*.f64 a x)

   1. Initial program 36.7%

      \[e^{a \cdot x} - 1 \]
   2. Step-by-step derivation

      1. expm1-define N/A

         \[\leadsto \mathsf{expm1}\left(a \cdot x\right) \]

      2. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{expm1.f64}\left(\left(a \cdot x\right)\right) \]

      3. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left(x + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot a\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(x + \frac{1}{2} \cdot \color{blue}{\left({x}^{2} \cdot a\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(x + \frac{1}{2} \cdot \left(a \cdot \color{blue}{{x}^{2}}\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(x + \left(\frac{1}{2} \cdot a\right) \cdot \color{blue}{{x}^{2}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(x + \left(\frac{1}{2} \cdot a\right) \cdot \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(x + \left(\left(\frac{1}{2} \cdot a\right) \cdot x\right) \cdot \color{blue}{x}\right)\right) \]

      9. distribute-rgt1-in N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(\left(\left(\frac{1}{2} \cdot a\right) \cdot x + 1\right) \cdot \color{blue}{x}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(a, \left(x \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot x + 1\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot x + 1\right)}\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \left(1 + \color{blue}{\left(\frac{1}{2} \cdot a\right) \cdot x}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\left(\frac{1}{2} \cdot a\right) \cdot x\right)}\right)\right)\right) \]

      14. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \color{blue}{\left(a \cdot x\right)}\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(a \cdot x\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(a \cdot x\right), \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 98.8%

          \[\leadsto \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x\right), \frac{1}{2}\right)\right)\right)\right) \]

   7. Simplified 98.8%

      \[\leadsto \color{blue}{a \cdot \left(x \cdot \left(1 + \left(a \cdot x\right) \cdot 0.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -10:\\ \;\;\;\;\frac{1}{1 - a \cdot x} + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(1 + \left(a \cdot x\right) \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -0.0001:\\ \;\;\;\;\frac{1}{1 - a \cdot x} + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -0.0001) (+ (/ 1.0 (- 1.0 (* a x))) -1.0) (* a x)))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -0.0001) {
		tmp = (1.0 / (1.0 - (a * x))) + -1.0;
	} else {
		tmp = a * x;
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((a * x) <= (-0.0001d0)) then
        tmp = (1.0d0 / (1.0d0 - (a * x))) + (-1.0d0)
    else
        tmp = a * x
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -0.0001) {
		tmp = (1.0 / (1.0 - (a * x))) + -1.0;
	} else {
		tmp = a * x;
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -0.0001:
		tmp = (1.0 / (1.0 - (a * x))) + -1.0
	else:
		tmp = a * x
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -0.0001)
		tmp = Float64(Float64(1.0 / Float64(1.0 - Float64(a * x))) + -1.0);
	else
		tmp = Float64(a * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -0.0001)
		tmp = (1.0 / (1.0 - (a * x))) + -1.0;
	else
		tmp = a * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -0.0001], N[(N[(1.0 / N[(1.0 - N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(a * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -1.00000000000000005e-4

   1. Initial program 99.7%

      \[e^{a \cdot x} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + a \cdot x\right)}, 1\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(a \cdot x + 1\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(a \cdot x\right), 1\right), 1\right) \]

      3. *-lowering-*.f64 6.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), 1\right), 1\right) \]

   5. Simplified 6.0%

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

      1. flip-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1}{a \cdot x - 1}\right), 1\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\frac{a \cdot x - 1}{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1}}\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{a \cdot x - 1}{\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1}\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(a \cdot x - 1\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(a \cdot x + \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a \cdot x\right), \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), \left(\mathsf{neg}\left(1\right)\right)\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1 \cdot 1\right)\right)\right), 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) - 1\right)\right)\right), 1\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\left(\left(x \cdot a\right) \cdot \left(a \cdot x\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\left(x \cdot \left(a \cdot \left(a \cdot x\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(a \cdot \left(a \cdot x\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \left(a \cdot x\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 1\right) \]

      17. metadata-eval 5.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), -1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, \mathsf{*.f64}\left(a, x\right)\right)\right), -1\right)\right)\right), 1\right) \]

   7. Applied egg-rr 5.0%

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

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(1 + -1 \cdot \left(a \cdot x\right)\right)}\right), 1\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \left(\mathsf{neg}\left(a \cdot x\right)\right)\right)\right), 1\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(1 - a \cdot x\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \left(a \cdot x\right)\right)\right), 1\right) \]

      4. *-lowering-*.f64 96.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), 1\right) \]

   10. Simplified 96.3%

       \[\leadsto \frac{1}{\color{blue}{1 - a \cdot x}} - 1 \]

   if -1.00000000000000005e-4 < (*.f64 a x)

   1. Initial program 36.1%

      \[e^{a \cdot x} - 1 \]
   2. Step-by-step derivation

      1. expm1-define N/A

         \[\leadsto \mathsf{expm1}\left(a \cdot x\right) \]

      2. expm1-lowering-expm1.f64 N/A

         \[\leadsto \mathsf{expm1.f64}\left(\left(a \cdot x\right)\right) \]

      3. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, x\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{a \cdot x} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 98.1%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{x}\right) \]

   7. Simplified 98.1%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -0.0001:\\ \;\;\;\;\frac{1}{1 - a \cdot x} + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.2% accurate, 35.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, x):
	return a * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.9%

   \[e^{a \cdot x} - 1 \]
2. Step-by-step derivation

   1. expm1-define N/A

      \[\leadsto \mathsf{expm1}\left(a \cdot x\right) \]

   2. expm1-lowering-expm1.f64 N/A

      \[\leadsto \mathsf{expm1.f64}\left(\left(a \cdot x\right)\right) \]

   3. *-lowering-*.f64 100.0%

      \[\leadsto \mathsf{expm1.f64}\left(\mathsf{*.f64}\left(a, x\right)\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]
4. Add Preprocessing

5. Taylor expanded in a around 0

   \[\leadsto \color{blue}{a \cdot x} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 66.4%

      \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{x}\right) \]

7. Simplified 66.4%

   \[\leadsto \color{blue}{a \cdot x} \]
8. Add Preprocessing

Alternative 10: 19.3% accurate, 105.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a x) :precision binary64 0.0)

C

double code(double a, double x) {
	return 0.0;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double a, double x) {
	return 0.0;
}

Python

def code(a, x):
	return 0.0

Julia

function code(a, x)
	return 0.0
end

MATLAB

function tmp = code(a, x)
	tmp = 0.0;
end

Wolfram

code[a_, x_] := 0.0

Derivation

1. Initial program 57.9%

   \[e^{a \cdot x} - 1 \]
2. Add Preprocessing

3. Taylor expanded in a around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, 1\right) \]
4. Step-by-step derivation

5. Simplified 22.5%

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

   1. metadata-eval 22.5%

      \[\leadsto 0 \]

7. Applied egg-rr 22.5%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double x) {
	return expm1((a * x));
}

Java

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

Python

def code(a, x):
	return math.expm1((a * x))

Julia

function code(a, x)
	return expm1(Float64(a * x))
end

Wolfram

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

Reproduce

herbie shell --seed 2024170 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :pre (> 710.0 (* a x))

  :alt
  (! :herbie-platform default (expm1 (* a x)))

  (- (exp (* a x)) 1.0))