expax (section 3.5)

Percentage Accurate: 53.9% → 100.0%

Time: 11.1s

Alternatives: 11

Speedup: 10.5×

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: 53.9% 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 50.9%

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

   1. accelerator-lowering-expm1.f64 N/A

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

   2. *-lowering-*.f64 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 2: 99.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -100:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + a \cdot \left(\left(a \cdot x\right) \cdot \left(x \cdot \left(0.5 + \left(a \cdot x\right) \cdot \left(0.16666666666666666 + a \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -100.0)
   -1.0
   (+
    (* a x)
    (*
     a
     (*
      (* a x)
      (*
       x
       (+
        0.5
        (*
         (* a x)
         (+ 0.16666666666666666 (* a (* x 0.041666666666666664)))))))))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -100.0) {
		tmp = -1.0;
	} else {
		tmp = (a * x) + (a * ((a * x) * (x * (0.5 + ((a * x) * (0.16666666666666666 + (a * (x * 0.041666666666666664))))))));
	}
	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) <= (-100.0d0)) then
        tmp = -1.0d0
    else
        tmp = (a * x) + (a * ((a * x) * (x * (0.5d0 + ((a * x) * (0.16666666666666666d0 + (a * (x * 0.041666666666666664d0))))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -100.0) {
		tmp = -1.0;
	} else {
		tmp = (a * x) + (a * ((a * x) * (x * (0.5 + ((a * x) * (0.16666666666666666 + (a * (x * 0.041666666666666664))))))));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -100.0:
		tmp = -1.0
	else:
		tmp = (a * x) + (a * ((a * x) * (x * (0.5 + ((a * x) * (0.16666666666666666 + (a * (x * 0.041666666666666664))))))))
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -100.0)
		tmp = -1.0;
	else
		tmp = Float64(Float64(a * x) + Float64(a * Float64(Float64(a * x) * Float64(x * Float64(0.5 + Float64(Float64(a * x) * Float64(0.16666666666666666 + Float64(a * Float64(x * 0.041666666666666664)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -100.0)
		tmp = -1.0;
	else
		tmp = (a * x) + (a * ((a * x) * (x * (0.5 + ((a * x) * (0.16666666666666666 + (a * (x * 0.041666666666666664))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -100.0], -1.0, N[(N[(a * x), $MachinePrecision] + N[(a * N[(N[(a * x), $MachinePrecision] * N[(x * N[(0.5 + N[(N[(a * x), $MachinePrecision] * N[(0.16666666666666666 + N[(a * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -100

   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 5.3%

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

   5. Simplified 5.3%

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

      1. flip3-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(a \cdot x\right)}^{3} + {1}^{3}}{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}\right), 1\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}{{\left(a \cdot x\right)}^{3} + {1}^{3}}}\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}{{\left(a \cdot x\right)}^{3} + {1}^{3}}\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a \cdot \left(x \cdot \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \left(x \cdot \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 - a \cdot x\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \left(a \cdot x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + 1\right)\right)\right), 1\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 + {\left(a \cdot x\right)}^{3}\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{+.f64}\left(1, \left({\left(a \cdot x\right)}^{3}\right)\right)\right)\right), 1\right) \]

      17. cube-mult N/A

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

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

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

   7. Applied egg-rr 2.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(1 + x \cdot \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)}\right), 1\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(x \cdot \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)\right)\right), 1\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) + -1 \cdot a\right)\right)\right)\right), 1\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) + \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right)\right), 1\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) - a\right)\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right), a\right)\right)\right)\right), 1\right) \]

   10. Simplified 98.9%

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

   11. Taylor expanded in x around inf

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

   13. Simplified 100.0%

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

   if -100 < (*.f64 a x)

   1. Initial program 27.3%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{a \cdot \left(x + a \cdot \left(\frac{1}{2} \cdot {x}^{2} + a \cdot \left(\frac{1}{24} \cdot \left(a \cdot {x}^{4}\right) + \frac{1}{6} \cdot {x}^{3}\right)\right)\right)} \]

   5. Simplified 99.0%

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

      1. distribute-rgt-in N/A

         \[\leadsto a \cdot \left(1 \cdot x + \color{blue}{\left(a \cdot \left(x \cdot \left(\frac{1}{2} + \left(a \cdot x\right) \cdot \left(\frac{1}{6} + \left(a \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right) \cdot x}\right) \]

      2. *-lft-identity N/A

         \[\leadsto a \cdot \left(x + \color{blue}{\left(a \cdot \left(x \cdot \left(\frac{1}{2} + \left(a \cdot x\right) \cdot \left(\frac{1}{6} + \left(a \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right)} \cdot x\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto x \cdot a + \color{blue}{\left(\left(a \cdot \left(x \cdot \left(\frac{1}{2} + \left(a \cdot x\right) \cdot \left(\frac{1}{6} + \left(a \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right) \cdot x\right) \cdot a} \]

      4. *-commutative N/A

         \[\leadsto a \cdot x + \color{blue}{\left(\left(a \cdot \left(x \cdot \left(\frac{1}{2} + \left(a \cdot x\right) \cdot \left(\frac{1}{6} + \left(a \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right) \cdot x\right)} \cdot a \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot x\right), \color{blue}{\left(\left(\left(a \cdot \left(x \cdot \left(\frac{1}{2} + \left(a \cdot x\right) \cdot \left(\frac{1}{6} + \left(a \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right) \cdot x\right) \cdot a\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), \left(\color{blue}{\left(\left(a \cdot \left(x \cdot \left(\frac{1}{2} + \left(a \cdot x\right) \cdot \left(\frac{1}{6} + \left(a \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right) \cdot x\right)} \cdot a\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, x\right), \mathsf{*.f64}\left(\left(\left(a \cdot \left(x \cdot \left(\frac{1}{2} + \left(a \cdot x\right) \cdot \left(\frac{1}{6} + \left(a \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right) \cdot x\right), \color{blue}{a}\right)\right) \]

   7. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{a \cdot x + \left(\left(a \cdot x\right) \cdot \left(x \cdot \left(0.5 + \left(a \cdot x\right) \cdot \left(0.16666666666666666 + a \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)\right)\right) \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -100:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + a \cdot \left(\left(a \cdot x\right) \cdot \left(x \cdot \left(0.5 + \left(a \cdot x\right) \cdot \left(0.16666666666666666 + a \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -100:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x + \left(a \cdot x\right) \cdot \left(x \cdot \left(0.5 + \left(a \cdot x\right) \cdot \left(0.16666666666666666 + a \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -100.0)
   -1.0
   (*
    a
    (+
     x
     (*
      (* a x)
      (*
       x
       (+
        0.5
        (*
         (* a x)
         (+ 0.16666666666666666 (* a (* x 0.041666666666666664)))))))))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -100.0) {
		tmp = -1.0;
	} else {
		tmp = a * (x + ((a * x) * (x * (0.5 + ((a * x) * (0.16666666666666666 + (a * (x * 0.041666666666666664))))))));
	}
	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) <= (-100.0d0)) then
        tmp = -1.0d0
    else
        tmp = a * (x + ((a * x) * (x * (0.5d0 + ((a * x) * (0.16666666666666666d0 + (a * (x * 0.041666666666666664d0))))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -100.0) {
		tmp = -1.0;
	} else {
		tmp = a * (x + ((a * x) * (x * (0.5 + ((a * x) * (0.16666666666666666 + (a * (x * 0.041666666666666664))))))));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -100.0:
		tmp = -1.0
	else:
		tmp = a * (x + ((a * x) * (x * (0.5 + ((a * x) * (0.16666666666666666 + (a * (x * 0.041666666666666664))))))))
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -100.0)
		tmp = -1.0;
	else
		tmp = Float64(a * Float64(x + Float64(Float64(a * x) * Float64(x * Float64(0.5 + Float64(Float64(a * x) * Float64(0.16666666666666666 + Float64(a * Float64(x * 0.041666666666666664)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -100.0)
		tmp = -1.0;
	else
		tmp = a * (x + ((a * x) * (x * (0.5 + ((a * x) * (0.16666666666666666 + (a * (x * 0.041666666666666664))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -100.0], -1.0, N[(a * N[(x + N[(N[(a * x), $MachinePrecision] * N[(x * N[(0.5 + N[(N[(a * x), $MachinePrecision] * N[(0.16666666666666666 + N[(a * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -100

   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 5.3%

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

   5. Simplified 5.3%

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

      1. flip3-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(a \cdot x\right)}^{3} + {1}^{3}}{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}\right), 1\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}{{\left(a \cdot x\right)}^{3} + {1}^{3}}}\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}{{\left(a \cdot x\right)}^{3} + {1}^{3}}\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a \cdot \left(x \cdot \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \left(x \cdot \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 - a \cdot x\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \left(a \cdot x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + 1\right)\right)\right), 1\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 + {\left(a \cdot x\right)}^{3}\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{+.f64}\left(1, \left({\left(a \cdot x\right)}^{3}\right)\right)\right)\right), 1\right) \]

      17. cube-mult N/A

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

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

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

   7. Applied egg-rr 2.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(1 + x \cdot \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)}\right), 1\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(x \cdot \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)\right)\right), 1\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) + -1 \cdot a\right)\right)\right)\right), 1\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) + \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right)\right), 1\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) - a\right)\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right), a\right)\right)\right)\right), 1\right) \]

   10. Simplified 98.9%

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

   11. Taylor expanded in x around inf

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

   13. Simplified 100.0%

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

   if -100 < (*.f64 a x)

   1. Initial program 27.3%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{a \cdot \left(x + a \cdot \left(\frac{1}{2} \cdot {x}^{2} + a \cdot \left(\frac{1}{24} \cdot \left(a \cdot {x}^{4}\right) + \frac{1}{6} \cdot {x}^{3}\right)\right)\right)} \]

   5. Simplified 99.0%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(x \cdot \left(a \cdot \left(x \cdot \left(\frac{1}{2} + \left(a \cdot x\right) \cdot \left(\frac{1}{6} + \left(a \cdot x\right) \cdot \frac{1}{24}\right)\right)\right) + \color{blue}{1}\right)\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(\left(a \cdot \left(x \cdot \left(\frac{1}{2} + \left(a \cdot x\right) \cdot \left(\frac{1}{6} + \left(a \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right) \cdot x + \color{blue}{1 \cdot x}\right)\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(\left(a \cdot \left(x \cdot \left(\frac{1}{2} + \left(a \cdot x\right) \cdot \left(\frac{1}{6} + \left(a \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right) \cdot x + x\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\left(a \cdot \left(x \cdot \left(\frac{1}{2} + \left(a \cdot x\right) \cdot \left(\frac{1}{6} + \left(a \cdot x\right) \cdot \frac{1}{24}\right)\right)\right)\right) \cdot x\right), \color{blue}{x}\right)\right) \]

   7. Applied egg-rr 99.0%

      \[\leadsto a \cdot \color{blue}{\left(\left(a \cdot x\right) \cdot \left(x \cdot \left(0.5 + \left(a \cdot x\right) \cdot \left(0.16666666666666666 + a \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)\right) + x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -100:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x + \left(a \cdot x\right) \cdot \left(x \cdot \left(0.5 + \left(a \cdot x\right) \cdot \left(0.16666666666666666 + a \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -100:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(1 + a \cdot \left(x \cdot \left(0.5 + \left(a \cdot x\right) \cdot \left(0.16666666666666666 + \left(a \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -100.0)
   -1.0
   (*
    a
    (*
     x
     (+
      1.0
      (*
       a
       (*
        x
        (+
         0.5
         (*
          (* a x)
          (+ 0.16666666666666666 (* (* a x) 0.041666666666666664)))))))))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -100.0) {
		tmp = -1.0;
	} else {
		tmp = a * (x * (1.0 + (a * (x * (0.5 + ((a * x) * (0.16666666666666666 + ((a * x) * 0.041666666666666664))))))));
	}
	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) <= (-100.0d0)) then
        tmp = -1.0d0
    else
        tmp = a * (x * (1.0d0 + (a * (x * (0.5d0 + ((a * x) * (0.16666666666666666d0 + ((a * x) * 0.041666666666666664d0))))))))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -100.0) {
		tmp = -1.0;
	} else {
		tmp = a * (x * (1.0 + (a * (x * (0.5 + ((a * x) * (0.16666666666666666 + ((a * x) * 0.041666666666666664))))))));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -100.0:
		tmp = -1.0
	else:
		tmp = a * (x * (1.0 + (a * (x * (0.5 + ((a * x) * (0.16666666666666666 + ((a * x) * 0.041666666666666664))))))))
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -100.0)
		tmp = -1.0;
	else
		tmp = Float64(a * Float64(x * Float64(1.0 + Float64(a * Float64(x * Float64(0.5 + Float64(Float64(a * x) * Float64(0.16666666666666666 + Float64(Float64(a * x) * 0.041666666666666664)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -100.0)
		tmp = -1.0;
	else
		tmp = a * (x * (1.0 + (a * (x * (0.5 + ((a * x) * (0.16666666666666666 + ((a * x) * 0.041666666666666664))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -100.0], -1.0, N[(a * N[(x * N[(1.0 + N[(a * N[(x * N[(0.5 + N[(N[(a * x), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(a * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -100

   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 5.3%

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

   5. Simplified 5.3%

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

      1. flip3-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(a \cdot x\right)}^{3} + {1}^{3}}{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}\right), 1\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}{{\left(a \cdot x\right)}^{3} + {1}^{3}}}\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}{{\left(a \cdot x\right)}^{3} + {1}^{3}}\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a \cdot \left(x \cdot \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \left(x \cdot \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 - a \cdot x\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \left(a \cdot x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + 1\right)\right)\right), 1\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 + {\left(a \cdot x\right)}^{3}\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{+.f64}\left(1, \left({\left(a \cdot x\right)}^{3}\right)\right)\right)\right), 1\right) \]

      17. cube-mult N/A

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

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

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

   7. Applied egg-rr 2.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(1 + x \cdot \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)}\right), 1\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(x \cdot \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)\right)\right), 1\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) + -1 \cdot a\right)\right)\right)\right), 1\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) + \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right)\right), 1\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) - a\right)\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right), a\right)\right)\right)\right), 1\right) \]

   10. Simplified 98.9%

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

   11. Taylor expanded in x around inf

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

   13. Simplified 100.0%

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

   if -100 < (*.f64 a x)

   1. Initial program 27.3%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{a \cdot \left(x + a \cdot \left(\frac{1}{2} \cdot {x}^{2} + a \cdot \left(\frac{1}{24} \cdot \left(a \cdot {x}^{4}\right) + \frac{1}{6} \cdot {x}^{3}\right)\right)\right)} \]

   5. Simplified 99.0%

      \[\leadsto \color{blue}{a \cdot \left(x \cdot \left(1 + a \cdot \left(x \cdot \left(0.5 + \left(a \cdot x\right) \cdot \left(0.16666666666666666 + \left(a \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -100:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(a \cdot x\right) \cdot \left(0.5 + x \cdot \left(a \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -100.0)
   -1.0
   (* (* a x) (+ 1.0 (* (* a x) (+ 0.5 (* x (* a 0.16666666666666666))))))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -100.0) {
		tmp = -1.0;
	} else {
		tmp = (a * x) * (1.0 + ((a * x) * (0.5 + (x * (a * 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) <= (-100.0d0)) then
        tmp = -1.0d0
    else
        tmp = (a * x) * (1.0d0 + ((a * x) * (0.5d0 + (x * (a * 0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -100.0) {
		tmp = -1.0;
	} else {
		tmp = (a * x) * (1.0 + ((a * x) * (0.5 + (x * (a * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -100.0:
		tmp = -1.0
	else:
		tmp = (a * x) * (1.0 + ((a * x) * (0.5 + (x * (a * 0.16666666666666666)))))
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -100.0)
		tmp = -1.0;
	else
		tmp = Float64(Float64(a * x) * Float64(1.0 + Float64(Float64(a * x) * Float64(0.5 + Float64(x * Float64(a * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -100.0)
		tmp = -1.0;
	else
		tmp = (a * x) * (1.0 + ((a * x) * (0.5 + (x * (a * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -100.0], -1.0, N[(N[(a * x), $MachinePrecision] * N[(1.0 + N[(N[(a * x), $MachinePrecision] * N[(0.5 + N[(x * N[(a * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -100

   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 5.3%

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

   5. Simplified 5.3%

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

      1. flip3-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(a \cdot x\right)}^{3} + {1}^{3}}{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}\right), 1\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}{{\left(a \cdot x\right)}^{3} + {1}^{3}}}\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}{{\left(a \cdot x\right)}^{3} + {1}^{3}}\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a \cdot \left(x \cdot \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \left(x \cdot \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 - a \cdot x\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \left(a \cdot x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + 1\right)\right)\right), 1\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 + {\left(a \cdot x\right)}^{3}\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{+.f64}\left(1, \left({\left(a \cdot x\right)}^{3}\right)\right)\right)\right), 1\right) \]

      17. cube-mult N/A

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

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

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

   7. Applied egg-rr 2.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(1 + x \cdot \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)}\right), 1\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(x \cdot \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)\right)\right), 1\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) + -1 \cdot a\right)\right)\right)\right), 1\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) + \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right)\right), 1\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) - a\right)\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right), a\right)\right)\right)\right), 1\right) \]

   10. Simplified 98.9%

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

   11. Taylor expanded in x around inf

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

   13. Simplified 100.0%

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

   if -100 < (*.f64 a x)

   1. Initial program 27.3%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{a \cdot \left(x + a \cdot \left(\frac{1}{6} \cdot \left(a \cdot {x}^{3}\right) + \frac{1}{2} \cdot {x}^{2}\right)\right)} \]
   4. 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

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

   5. Simplified 98.8%

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

Alternative 6: 98.9% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -100:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + a \cdot \left(x \cdot \left(a \cdot \frac{x}{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -100.0) -1.0 (+ (* a x) (* a (* x (* a (/ x 2.0)))))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -100.0) {
		tmp = -1.0;
	} else {
		tmp = (a * x) + (a * (x * (a * (x / 2.0))));
	}
	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) <= (-100.0d0)) then
        tmp = -1.0d0
    else
        tmp = (a * x) + (a * (x * (a * (x / 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -100.0) {
		tmp = -1.0;
	} else {
		tmp = (a * x) + (a * (x * (a * (x / 2.0))));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -100.0:
		tmp = -1.0
	else:
		tmp = (a * x) + (a * (x * (a * (x / 2.0))))
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -100.0)
		tmp = -1.0;
	else
		tmp = Float64(Float64(a * x) + Float64(a * Float64(x * Float64(a * Float64(x / 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -100.0)
		tmp = -1.0;
	else
		tmp = (a * x) + (a * (x * (a * (x / 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -100.0], -1.0, N[(N[(a * x), $MachinePrecision] + N[(a * N[(x * N[(a * N[(x / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -100

   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 5.3%

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

   5. Simplified 5.3%

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

      1. flip3-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(a \cdot x\right)}^{3} + {1}^{3}}{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}\right), 1\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}{{\left(a \cdot x\right)}^{3} + {1}^{3}}}\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}{{\left(a \cdot x\right)}^{3} + {1}^{3}}\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a \cdot \left(x \cdot \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \left(x \cdot \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 - a \cdot x\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \left(a \cdot x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + 1\right)\right)\right), 1\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 + {\left(a \cdot x\right)}^{3}\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{+.f64}\left(1, \left({\left(a \cdot x\right)}^{3}\right)\right)\right)\right), 1\right) \]

      17. cube-mult N/A

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

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

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

   7. Applied egg-rr 2.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(1 + x \cdot \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)}\right), 1\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(x \cdot \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)\right)\right), 1\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) + -1 \cdot a\right)\right)\right)\right), 1\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) + \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right)\right), 1\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) - a\right)\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right), a\right)\right)\right)\right), 1\right) \]

   10. Simplified 98.9%

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

   11. Taylor expanded in x around inf

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

   13. Simplified 100.0%

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

   if -100 < (*.f64 a x)

   1. Initial program 27.3%

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

   3. Taylor expanded in a around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. +-commutative N/A

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

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

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

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

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

      17. *-lowering-*.f64 98.5%

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

   5. Simplified 98.5%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. associate-*l* N/A

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

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

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

      11. metadata-eval N/A

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

      12. div-inv N/A

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

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

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

      14. *-lowering-*.f64 98.5%

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

   7. Applied egg-rr 98.5%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -100:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + a \cdot \left(x \cdot \left(a \cdot \frac{x}{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.9% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -100:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(a \cdot x\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -100.0) -1.0 (* (* a x) (+ 1.0 (* (* a x) 0.5)))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -100.0) {
		tmp = -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) <= (-100.0d0)) then
        tmp = -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) <= -100.0) {
		tmp = -1.0;
	} else {
		tmp = (a * x) * (1.0 + ((a * x) * 0.5));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -100.0:
		tmp = -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) <= -100.0)
		tmp = -1.0;
	else
		tmp = Float64(Float64(a * 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) <= -100.0)
		tmp = -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], -100.0], -1.0, N[(N[(a * x), $MachinePrecision] * N[(1.0 + N[(N[(a * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -100

   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 5.3%

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

   5. Simplified 5.3%

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

      1. flip3-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(a \cdot x\right)}^{3} + {1}^{3}}{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}\right), 1\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}{{\left(a \cdot x\right)}^{3} + {1}^{3}}}\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}{{\left(a \cdot x\right)}^{3} + {1}^{3}}\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a \cdot \left(x \cdot \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \left(x \cdot \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 - a \cdot x\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \left(a \cdot x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + 1\right)\right)\right), 1\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 + {\left(a \cdot x\right)}^{3}\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{+.f64}\left(1, \left({\left(a \cdot x\right)}^{3}\right)\right)\right)\right), 1\right) \]

      17. cube-mult N/A

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

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

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

   7. Applied egg-rr 2.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(1 + x \cdot \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)}\right), 1\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(x \cdot \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)\right)\right), 1\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) + -1 \cdot a\right)\right)\right)\right), 1\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) + \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right)\right), 1\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) - a\right)\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right), a\right)\right)\right)\right), 1\right) \]

   10. Simplified 98.9%

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

   11. Taylor expanded in x around inf

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

   13. Simplified 100.0%

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

   if -100 < (*.f64 a x)

   1. Initial program 27.3%

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

   3. Taylor expanded in a around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. +-commutative N/A

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

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

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

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

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

      17. *-lowering-*.f64 98.5%

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

   5. Simplified 98.5%

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

Alternative 8: 98.9% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -100:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(1 + a \cdot \left(x \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -100.0) -1.0 (* a (* x (+ 1.0 (* a (* x 0.5)))))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -100.0) {
		tmp = -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) <= (-100.0d0)) then
        tmp = -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) <= -100.0) {
		tmp = -1.0;
	} else {
		tmp = a * (x * (1.0 + (a * (x * 0.5))));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -100.0:
		tmp = -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) <= -100.0)
		tmp = -1.0;
	else
		tmp = Float64(a * Float64(x * 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) <= -100.0)
		tmp = -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], -100.0], -1.0, N[(a * N[(x * 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) < -100

   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 5.3%

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

   5. Simplified 5.3%

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

      1. flip3-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(a \cdot x\right)}^{3} + {1}^{3}}{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}\right), 1\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}{{\left(a \cdot x\right)}^{3} + {1}^{3}}}\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}{{\left(a \cdot x\right)}^{3} + {1}^{3}}\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a \cdot \left(x \cdot \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \left(x \cdot \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 - a \cdot x\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \left(a \cdot x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + 1\right)\right)\right), 1\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 + {\left(a \cdot x\right)}^{3}\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{+.f64}\left(1, \left({\left(a \cdot x\right)}^{3}\right)\right)\right)\right), 1\right) \]

      17. cube-mult N/A

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

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

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

   7. Applied egg-rr 2.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(1 + x \cdot \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)}\right), 1\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(x \cdot \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)\right)\right), 1\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) + -1 \cdot a\right)\right)\right)\right), 1\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) + \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right)\right), 1\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) - a\right)\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right), a\right)\right)\right)\right), 1\right) \]

   10. Simplified 98.9%

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

   11. Taylor expanded in x around inf

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

   13. Simplified 100.0%

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

   if -100 < (*.f64 a x)

   1. Initial program 27.3%

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

      1. accelerator-lowering-expm1.f64 N/A

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

      2. *-lowering-*.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   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. distribute-lft-in N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft-in N/A

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

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

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-lft-identity N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*l* N/A

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

      12. distribute-rgt-in N/A

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

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

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

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

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

      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. associate-*r* N/A

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

      17. *-commutative N/A

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

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

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

      19. *-commutative N/A

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

      20. *-lowering-*.f64 98.5%

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

   7. Simplified 98.5%

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

Alternative 9: 98.2% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -100:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (a x) :precision binary64 (if (<= (* a x) -100.0) -1.0 (* a x)))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -100.0) {
		tmp = -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) <= (-100.0d0)) then
        tmp = -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) <= -100.0) {
		tmp = -1.0;
	} else {
		tmp = a * x;
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -100.0:
		tmp = -1.0
	else:
		tmp = a * x
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -100.0)
		tmp = -1.0;
	else
		tmp = Float64(a * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -100.0)
		tmp = -1.0;
	else
		tmp = a * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -100.0], -1.0, N[(a * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -100

   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 5.3%

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

   5. Simplified 5.3%

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

      1. flip3-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(a \cdot x\right)}^{3} + {1}^{3}}{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}\right), 1\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}{{\left(a \cdot x\right)}^{3} + {1}^{3}}}\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}{{\left(a \cdot x\right)}^{3} + {1}^{3}}\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a \cdot \left(x \cdot \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \left(x \cdot \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 - a \cdot x\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \left(a \cdot x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + 1\right)\right)\right), 1\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 + {\left(a \cdot x\right)}^{3}\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{+.f64}\left(1, \left({\left(a \cdot x\right)}^{3}\right)\right)\right)\right), 1\right) \]

      17. cube-mult N/A

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

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

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

   7. Applied egg-rr 2.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(1 + x \cdot \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)}\right), 1\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(x \cdot \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)\right)\right), 1\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) + -1 \cdot a\right)\right)\right)\right), 1\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) + \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right)\right), 1\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) - a\right)\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right), a\right)\right)\right)\right), 1\right) \]

   10. Simplified 98.9%

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

   11. Taylor expanded in x around inf

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

   13. Simplified 100.0%

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

   if -100 < (*.f64 a x)

   1. Initial program 27.3%

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

   3. Taylor expanded in a around 0

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

      1. *-lowering-*.f64 97.7%

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

   5. Simplified 97.7%

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

Alternative 10: 32.9% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-146}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (a x) :precision binary64 (if (<= x 1.85e-146) 0.0 -1.0))

C

double code(double a, double x) {
	double tmp;
	if (x <= 1.85e-146) {
		tmp = 0.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.85d-146) then
        tmp = 0.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if (x <= 1.85e-146) {
		tmp = 0.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if x <= 1.85e-146:
		tmp = 0.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (x <= 1.85e-146)
		tmp = 0.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if (x <= 1.85e-146)
		tmp = 0.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[x, 1.85e-146], 0.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.84999999999999993e-146

   1. Initial program 49.7%

      \[e^{a \cdot x} - 1 \]
   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 21.9%

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

      1. metadata-eval 21.9%

         \[\leadsto 0 \]

   7. Applied egg-rr 21.9%

      \[\leadsto \color{blue}{0} \]

   if 1.84999999999999993e-146 < x

   1. Initial program 53.6%

      \[e^{a \cdot x} - 1 \]
   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 8.8%

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

   5. Simplified 8.8%

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

      1. flip3-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(a \cdot x\right)}^{3} + {1}^{3}}{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}\right), 1\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}{{\left(a \cdot x\right)}^{3} + {1}^{3}}}\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}{{\left(a \cdot x\right)}^{3} + {1}^{3}}\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a \cdot \left(x \cdot \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \left(x \cdot \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      11. *-rgt-identity N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 - a \cdot x\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \left(a \cdot x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + 1\right)\right)\right), 1\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 + {\left(a \cdot x\right)}^{3}\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{+.f64}\left(1, \left({\left(a \cdot x\right)}^{3}\right)\right)\right)\right), 1\right) \]

      17. cube-mult N/A

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

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

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

   7. Applied egg-rr 7.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(1 + x \cdot \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)}\right), 1\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(x \cdot \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)\right)\right), 1\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) + -1 \cdot a\right)\right)\right)\right), 1\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) + \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right)\right), 1\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) - a\right)\right)\right)\right), 1\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right), a\right)\right)\right)\right), 1\right) \]

   10. Simplified 53.2%

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

   11. Taylor expanded in x around inf

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

   13. Simplified 48.8%

       \[\leadsto \color{blue}{-1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 35.7% accurate, 105.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, x):
	return -1.0

Julia

function code(a, x)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, x_] := -1.0

Derivation

1. Initial program 50.9%

   \[e^{a \cdot x} - 1 \]
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 19.1%

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

5. Simplified 19.1%

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

   1. flip3-+ N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{{\left(a \cdot x\right)}^{3} + {1}^{3}}{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}\right), 1\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}{{\left(a \cdot x\right)}^{3} + {1}^{3}}}\right), 1\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)}{{\left(a \cdot x\right)}^{3} + {1}^{3}}\right)\right), 1\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right) + \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(a \cdot \left(x \cdot \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \left(x \cdot \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \left(a \cdot x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 \cdot 1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 - \left(a \cdot x\right) \cdot 1\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

   11. *-rgt-identity N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 - a \cdot x\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \left(a \cdot x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

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

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + {1}^{3}\right)\right)\right), 1\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left({\left(a \cdot x\right)}^{3} + 1\right)\right)\right), 1\right) \]

   15. +-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \left(1 + {\left(a \cdot x\right)}^{3}\right)\right)\right), 1\right) \]

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

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(a, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(a, x\right)\right)\right), \mathsf{+.f64}\left(1, \left({\left(a \cdot x\right)}^{3}\right)\right)\right)\right), 1\right) \]

   17. cube-mult N/A

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

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

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

7. Applied egg-rr 18.4%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(1 + x \cdot \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)}\right), 1\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(x \cdot \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)\right)\right), 1\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 \cdot a + x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right)\right)\right)\right), 1\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) + -1 \cdot a\right)\right)\right)\right), 1\right) \]

   4. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) + \left(\mathsf{neg}\left(a\right)\right)\right)\right)\right)\right), 1\right) \]

   5. unsub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right) - a\right)\right)\right)\right), 1\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(x \cdot \left(-1 \cdot \left({a}^{3} \cdot x\right) + {a}^{2}\right)\right), a\right)\right)\right)\right), 1\right) \]

10. Simplified 49.3%

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

11. Taylor expanded in x around inf

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

13. Simplified 34.9%

    \[\leadsto \color{blue}{-1} \]
14. 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 2024192 
(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))