expax (section 3.5)

Percentage Accurate: 53.5% → 100.0%

Time: 13.1s

Alternatives: 7

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: 53.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.6%

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

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

3. Simplified 100.0%

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

Alternative 2: 99.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -1000:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(1 + a \cdot \left(x \cdot \left(0.5 + \left(a \cdot x\right) \cdot \left(0.16666666666666666 + \left(a \cdot x\right) \cdot 0.041666666666666664\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -1000.0)
   -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) <= -1000.0) {
		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) <= (-1000.0d0)) 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) <= -1000.0) {
		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) <= -1000.0:
		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) <= -1000.0)
		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) <= -1000.0)
		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], -1000.0], -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) < -1e3

   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 4.9%

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

   5. Simplified 4.9%

      \[\leadsto \color{blue}{\left(a \cdot x + 1\right)} - 1 \]
   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 2.1%

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

   8. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. *-lowering-*.f64 98.8%

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

   10. Simplified 98.8%

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

   11. Taylor expanded in a around inf

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

   13. Simplified 100.0%

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

   if -1e3 < (*.f64 a x)

   1. Initial program 26.5%

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

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0

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

   7. Simplified 99.1%

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

Alternative 3: 99.1% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -1000:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(a \cdot x\right) \cdot \left(0.5 + x \cdot \left(a \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -1000.0)
   -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) <= -1000.0) {
		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) <= (-1000.0d0)) 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) <= -1000.0) {
		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) <= -1000.0:
		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) <= -1000.0)
		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) <= -1000.0)
		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], -1000.0], -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) < -1e3

   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 4.9%

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

   5. Simplified 4.9%

      \[\leadsto \color{blue}{\left(a \cdot x + 1\right)} - 1 \]
   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 2.1%

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

   8. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. *-lowering-*.f64 98.8%

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

   10. Simplified 98.8%

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

   11. Taylor expanded in a around inf

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

   13. Simplified 100.0%

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

   if -1e3 < (*.f64 a x)

   1. Initial program 26.5%

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

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. unpow3 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-rgt-out N/A

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

   7. Simplified 99.1%

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

Alternative 4: 98.8% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -1000:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(1 + \left(a \cdot x\right) \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -1000.0) -1.0 (* a (* x (+ 1.0 (* (* a x) 0.5))))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -1000.0) {
		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) <= (-1000.0d0)) 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) <= -1000.0) {
		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) <= -1000.0:
		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) <= -1000.0)
		tmp = -1.0;
	else
		tmp = Float64(a * Float64(x * Float64(1.0 + Float64(Float64(a * x) * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -1000.0)
		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], -1000.0], -1.0, N[(a * N[(x * N[(1.0 + N[(N[(a * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -1e3

   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 4.9%

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

   5. Simplified 4.9%

      \[\leadsto \color{blue}{\left(a \cdot x + 1\right)} - 1 \]
   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 2.1%

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

   8. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. *-lowering-*.f64 98.8%

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

   10. Simplified 98.8%

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

   11. Taylor expanded in a around inf

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

   13. Simplified 100.0%

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

   if -1e3 < (*.f64 a x)

   1. Initial program 26.5%

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

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-+l+ N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. unpow3 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. distribute-lft1-in N/A

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

      14. distribute-rgt-out N/A

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

   7. Simplified 99.1%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

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

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

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

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

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

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

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

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

      18. *-lowering-*.f64 99.1%

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

   9. Applied egg-rr 99.1%

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

   10. Taylor expanded in a around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. +-commutative N/A

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

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

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. unpow2 N/A

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

       9. associate-*r* N/A

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

       10. associate-*r* N/A

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

       11. *-lft-identity N/A

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

       12. distribute-rgt-out N/A

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

       13. +-commutative N/A

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

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

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

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

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

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

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

       17. *-lowering-*.f64 99.0%

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

   12. Simplified 99.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -1000:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(x \cdot \left(1 + \left(a \cdot x\right) \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.2% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -1000:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (a x) :precision binary64 (if (<= (* a x) -1000.0) -1.0 (* a x)))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -1000.0) {
		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) <= (-1000.0d0)) 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) <= -1000.0) {
		tmp = -1.0;
	} else {
		tmp = a * x;
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -1000.0:
		tmp = -1.0
	else:
		tmp = a * x
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -1000.0)
		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) <= -1000.0)
		tmp = -1.0;
	else
		tmp = a * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -1000.0], -1.0, N[(a * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -1e3

   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 4.9%

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

   5. Simplified 4.9%

      \[\leadsto \color{blue}{\left(a \cdot x + 1\right)} - 1 \]
   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 2.1%

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

   8. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. *-lowering-*.f64 98.8%

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

   10. Simplified 98.8%

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

   11. Taylor expanded in a around inf

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

   13. Simplified 100.0%

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

   if -1e3 < (*.f64 a x)

   1. Initial program 26.5%

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

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

   3. Simplified 100.0%

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

   5. Taylor expanded in a around 0

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

      1. *-lowering-*.f64 98.5%

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

   7. Simplified 98.5%

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

Alternative 6: 32.2% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.6 \cdot 10^{-95}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a x) :precision binary64 (if (<= a -5.6e-95) -1.0 0.0))

C

double code(double a, double x) {
	double tmp;
	if (a <= -5.6e-95) {
		tmp = -1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if (a <= (-5.6d-95)) then
        tmp = -1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if (a <= -5.6e-95) {
		tmp = -1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if a <= -5.6e-95:
		tmp = -1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (a <= -5.6e-95)
		tmp = -1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if (a <= -5.6e-95)
		tmp = -1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[a, -5.6e-95], -1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if a < -5.5999999999999998e-95

   1. Initial program 62.6%

      \[e^{a \cdot x} - 1 \]
   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 7.7%

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

   5. Simplified 7.7%

      \[\leadsto \color{blue}{\left(a \cdot x + 1\right)} - 1 \]
   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 6.2%

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

   8. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. *-lowering-*.f64 60.2%

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

   10. Simplified 60.2%

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

   11. Taylor expanded in a around inf

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

   13. Simplified 58.2%

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

   if -5.5999999999999998e-95 < a

   1. Initial program 46.2%

      \[e^{a \cdot x} - 1 \]
   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 21.7%

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

      1. metadata-eval 21.7%

         \[\leadsto 0 \]

   7. Applied egg-rr 21.7%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 35.3% 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 50.6%

   \[e^{a \cdot x} - 1 \]
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 18.8%

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

5. Simplified 18.8%

   \[\leadsto \color{blue}{\left(a \cdot x + 1\right)} - 1 \]
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 17.8%

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

8. Taylor expanded in a around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

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

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

   4. *-lowering-*.f64 49.5%

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

10. Simplified 49.5%

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

11. Taylor expanded in a around inf

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

13. Simplified 35.3%

    \[\leadsto \color{blue}{-1} \]
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 2024191 
(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))