expax (section 3.5)

Percentage Accurate: 53.7% → 99.9%

Time: 9.7s

Alternatives: 9

Speedup: 18.2×

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: 53.7% 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: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{{\left(e^{2}\right)}^{\left(0.5 \cdot \left(a \cdot x\right)\right)} + 1} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (/ (expm1 (* a (+ x x))) (+ (pow (exp 2.0) (* 0.5 (* a x))) 1.0)))

C

double code(double a, double x) {
	return expm1((a * (x + x))) / (pow(exp(2.0), (0.5 * (a * x))) + 1.0);
}

Java

public static double code(double a, double x) {
	return Math.expm1((a * (x + x))) / (Math.pow(Math.exp(2.0), (0.5 * (a * x))) + 1.0);
}

Python

def code(a, x):
	return math.expm1((a * (x + x))) / (math.pow(math.exp(2.0), (0.5 * (a * x))) + 1.0)

Julia

function code(a, x)
	return Float64(expm1(Float64(a * Float64(x + x))) / Float64((exp(2.0) ^ Float64(0.5 * Float64(a * x))) + 1.0))
end

Wolfram

code[a_, x_] := N[(N[(Exp[N[(a * N[(x + x), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision] / N[(N[Power[N[Exp[2.0], $MachinePrecision], N[(0.5 * N[(a * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.6%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. flip-- N/A

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

   4. lower-/.f64 N/A

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

   5. lift-exp.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. prod-exp N/A

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

   8. metadata-eval N/A

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

   9. lower-expm1.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(a \cdot x + a \cdot x\right)}}{e^{a \cdot x} + 1} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{a \cdot x} + a \cdot x\right)}{e^{a \cdot x} + 1} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{expm1}\left(a \cdot x + \color{blue}{a \cdot x}\right)}{e^{a \cdot x} + 1} \]

   12. distribute-lft-out N/A

       \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{a \cdot \left(x + x\right)}\right)}{e^{a \cdot x} + 1} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{a \cdot \left(x + x\right)}\right)}{e^{a \cdot x} + 1} \]

   14. lower-+.f64 N/A

       \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \color{blue}{\left(x + x\right)}\right)}{e^{a \cdot x} + 1} \]

   15. lower-+.f64 100.0

       \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{\color{blue}{e^{a \cdot x} + 1}} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{e^{a \cdot x} + 1}} \]
5. Step-by-step derivation

   1. exp-prod N/A

      \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{\color{blue}{{\left(e^{a}\right)}^{x}} + 1} \]

   2. sqr-pow N/A

      \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{\color{blue}{{\left(e^{a}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{a}\right)}^{\left(\frac{x}{2}\right)}} + 1} \]

   3. pow2 N/A

      \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{\color{blue}{{\left({\left(e^{a}\right)}^{\left(\frac{x}{2}\right)}\right)}^{2}} + 1} \]

   4. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{\color{blue}{{\left({\left(e^{a}\right)}^{\left(\frac{x}{2}\right)}\right)}^{2}} + 1} \]

   5. pow-exp N/A

      \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{{\color{blue}{\left(e^{a \cdot \frac{x}{2}}\right)}}^{2} + 1} \]

   6. lower-exp.f64 N/A

      \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{{\color{blue}{\left(e^{a \cdot \frac{x}{2}}\right)}}^{2} + 1} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{{\left(e^{\color{blue}{a \cdot \frac{x}{2}}}\right)}^{2} + 1} \]

   8. div-inv N/A

      \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{{\left(e^{a \cdot \color{blue}{\left(x \cdot \frac{1}{2}\right)}}\right)}^{2} + 1} \]

   9. metadata-eval N/A

      \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{{\left(e^{a \cdot \left(x \cdot \color{blue}{\frac{1}{2}}\right)}\right)}^{2} + 1} \]

   10. lower-*.f64 100.0

       \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{{\left(e^{a \cdot \color{blue}{\left(x \cdot 0.5\right)}}\right)}^{2} + 1} \]

6. Applied rewrites 100.0%

   \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{\color{blue}{{\left(e^{a \cdot \left(x \cdot 0.5\right)}\right)}^{2}} + 1} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{{\left(e^{a \cdot \color{blue}{\left(x \cdot \frac{1}{2}\right)}}\right)}^{2} + 1} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{{\left(e^{\color{blue}{a \cdot \left(x \cdot \frac{1}{2}\right)}}\right)}^{2} + 1} \]

   3. lift-exp.f64 N/A

      \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{{\color{blue}{\left(e^{a \cdot \left(x \cdot \frac{1}{2}\right)}\right)}}^{2} + 1} \]

   4. unpow1 N/A

      \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{{\color{blue}{\left({\left(e^{a \cdot \left(x \cdot \frac{1}{2}\right)}\right)}^{1}\right)}}^{2} + 1} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{{\left({\left(e^{a \cdot \left(x \cdot \frac{1}{2}\right)}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)}^{2} + 1} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{{\left({\left(e^{a \cdot \left(x \cdot \frac{1}{2}\right)}\right)}^{\color{blue}{1}}\right)}^{2} + 1} \]

   7. unpow1 N/A

      \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{{\color{blue}{\left(e^{a \cdot \left(x \cdot \frac{1}{2}\right)}\right)}}^{2} + 1} \]

   8. lift-exp.f64 N/A

      \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{{\color{blue}{\left(e^{a \cdot \left(x \cdot \frac{1}{2}\right)}\right)}}^{2} + 1} \]

   9. pow-exp N/A

      \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{\color{blue}{e^{\left(a \cdot \left(x \cdot \frac{1}{2}\right)\right) \cdot 2}} + 1} \]

   10. *-commutative N/A

       \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{e^{\color{blue}{2 \cdot \left(a \cdot \left(x \cdot \frac{1}{2}\right)\right)}} + 1} \]

   11. exp-prod N/A

       \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{\color{blue}{{\left(e^{2}\right)}^{\left(a \cdot \left(x \cdot \frac{1}{2}\right)\right)}} + 1} \]

   12. lower-pow.f64 N/A

       \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{\color{blue}{{\left(e^{2}\right)}^{\left(a \cdot \left(x \cdot \frac{1}{2}\right)\right)}} + 1} \]

   13. lower-exp.f64 100.0

       \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{{\color{blue}{\left(e^{2}\right)}}^{\left(a \cdot \left(x \cdot 0.5\right)\right)} + 1} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{{\left(e^{2}\right)}^{\color{blue}{\left(a \cdot \left(x \cdot \frac{1}{2}\right)\right)}} + 1} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{{\left(e^{2}\right)}^{\left(a \cdot \color{blue}{\left(x \cdot \frac{1}{2}\right)}\right)} + 1} \]

   16. associate-*r* N/A

       \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{{\left(e^{2}\right)}^{\color{blue}{\left(\left(a \cdot x\right) \cdot \frac{1}{2}\right)}} + 1} \]

   17. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{{\left(e^{2}\right)}^{\left(\color{blue}{\left(a \cdot x\right)} \cdot \frac{1}{2}\right)} + 1} \]

   18. *-commutative N/A

       \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{{\left(e^{2}\right)}^{\color{blue}{\left(\frac{1}{2} \cdot \left(a \cdot x\right)\right)}} + 1} \]

   19. lower-*.f64 100.0

       \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{{\left(e^{2}\right)}^{\color{blue}{\left(0.5 \cdot \left(a \cdot x\right)\right)}} + 1} \]

8. Applied rewrites 100.0%

   \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{\color{blue}{{\left(e^{2}\right)}^{\left(0.5 \cdot \left(a \cdot x\right)\right)}} + 1} \]
9. Add Preprocessing

Alternative 2: 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. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lower-expm1.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 3: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -20000:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(a, x, -1\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \left(a \cdot x\right), x \cdot \mathsf{fma}\left(a, x \cdot \mathsf{fma}\left(a, x \cdot 0.041666666666666664, 0.16666666666666666\right), 0.5\right), a \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -20000.0)
   (+ (/ -1.0 (fma a x -1.0)) -1.0)
   (fma
    (* a (* a x))
    (*
     x
     (fma a (* x (fma a (* x 0.041666666666666664) 0.16666666666666666)) 0.5))
    (* a x))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -20000.0) {
		tmp = (-1.0 / fma(a, x, -1.0)) + -1.0;
	} else {
		tmp = fma((a * (a * x)), (x * fma(a, (x * fma(a, (x * 0.041666666666666664), 0.16666666666666666)), 0.5)), (a * x));
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -20000.0)
		tmp = Float64(Float64(-1.0 / fma(a, x, -1.0)) + -1.0);
	else
		tmp = fma(Float64(a * Float64(a * x)), Float64(x * fma(a, Float64(x * fma(a, Float64(x * 0.041666666666666664), 0.16666666666666666)), 0.5)), Float64(a * x));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -20000.0], N[(N[(-1.0 / N[(a * x + -1.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(a * N[(a * x), $MachinePrecision]), $MachinePrecision] * N[(x * N[(a * N[(x * N[(a * N[(x * 0.041666666666666664), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + N[(a * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -2e4

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-*.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval N/A

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

      15. sub-neg N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 N/A

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

      18. metadata-eval 8.5

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

   7. Applied rewrites 8.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 98.4%

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

   if -2e4 < (*.f64 a x)

   1. Initial program 25.7%

      \[e^{a \cdot x} - 1 \]
   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. Applied rewrites 90.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. lift-*.f64 N/A

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

   7. Applied rewrites 99.1%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -20000:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(a, x, -1\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \left(a \cdot x\right), x \cdot \mathsf{fma}\left(a, x \cdot \mathsf{fma}\left(a, x \cdot 0.041666666666666664, 0.16666666666666666\right), 0.5\right), a \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -20000:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(a, x, -1\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(\left(a \cdot x\right) \cdot \mathsf{fma}\left(a, x \cdot \mathsf{fma}\left(a, x \cdot 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -20000.0)
   (+ (/ -1.0 (fma a x -1.0)) -1.0)
   (*
    a
    (fma
     (*
      (* a x)
      (fma a (* x (fma a (* x 0.041666666666666664) 0.16666666666666666)) 0.5))
     x
     x))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -20000.0) {
		tmp = (-1.0 / fma(a, x, -1.0)) + -1.0;
	} else {
		tmp = a * fma(((a * x) * fma(a, (x * fma(a, (x * 0.041666666666666664), 0.16666666666666666)), 0.5)), x, x);
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -20000.0)
		tmp = Float64(Float64(-1.0 / fma(a, x, -1.0)) + -1.0);
	else
		tmp = Float64(a * fma(Float64(Float64(a * x) * fma(a, Float64(x * fma(a, Float64(x * 0.041666666666666664), 0.16666666666666666)), 0.5)), x, x));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -20000.0], N[(N[(-1.0 / N[(a * x + -1.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(a * N[(N[(N[(a * x), $MachinePrecision] * N[(a * N[(x * N[(a * N[(x * 0.041666666666666664), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -2e4

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-*.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval N/A

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

      15. sub-neg N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 N/A

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

      18. metadata-eval 8.5

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

   7. Applied rewrites 8.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 98.4%

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

   if -2e4 < (*.f64 a x)

   1. Initial program 25.7%

      \[e^{a \cdot x} - 1 \]
   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. Applied rewrites 90.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

   7. Applied rewrites 99.1%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -20000:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(a, x, -1\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(\left(a \cdot x\right) \cdot \mathsf{fma}\left(a, x \cdot \mathsf{fma}\left(a, x \cdot 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -20000:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(a, x, -1\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(\left(a \cdot x\right) \cdot \mathsf{fma}\left(a, x \cdot 0.16666666666666666, 0.5\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -20000.0)
   (+ (/ -1.0 (fma a x -1.0)) -1.0)
   (* a (fma (* (* a x) (fma a (* x 0.16666666666666666) 0.5)) x x))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -20000.0) {
		tmp = (-1.0 / fma(a, x, -1.0)) + -1.0;
	} else {
		tmp = a * fma(((a * x) * fma(a, (x * 0.16666666666666666), 0.5)), x, x);
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -20000.0)
		tmp = Float64(Float64(-1.0 / fma(a, x, -1.0)) + -1.0);
	else
		tmp = Float64(a * fma(Float64(Float64(a * x) * fma(a, Float64(x * 0.16666666666666666), 0.5)), x, x));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -20000.0], N[(N[(-1.0 / N[(a * x + -1.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(a * N[(N[(N[(a * x), $MachinePrecision] * N[(a * N[(x * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -2e4

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-*.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval N/A

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

      15. sub-neg N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 N/A

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

      18. metadata-eval 8.5

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

   7. Applied rewrites 8.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 98.4%

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

   if -2e4 < (*.f64 a x)

   1. Initial program 25.7%

      \[e^{a \cdot x} - 1 \]
   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. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. associate-*r* N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 90.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 99.1

          \[\leadsto a \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(a, x \cdot 0.16666666666666666, 0.5\right) \cdot \left(a \cdot x\right)}, x, x\right) \]

   7. Applied rewrites 99.1%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -20000:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(a, x, -1\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(\left(a \cdot x\right) \cdot \mathsf{fma}\left(a, x \cdot 0.16666666666666666, 0.5\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -20000:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(a, x, -1\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(x, a \cdot \left(x \cdot 0.5\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -20000.0)
   (+ (/ -1.0 (fma a x -1.0)) -1.0)
   (* a (fma x (* a (* x 0.5)) x))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -20000.0) {
		tmp = (-1.0 / fma(a, x, -1.0)) + -1.0;
	} else {
		tmp = a * fma(x, (a * (x * 0.5)), x);
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -20000.0)
		tmp = Float64(Float64(-1.0 / fma(a, x, -1.0)) + -1.0);
	else
		tmp = Float64(a * fma(x, Float64(a * Float64(x * 0.5)), x));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -20000.0], N[(N[(-1.0 / N[(a * x + -1.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(a * N[(x * N[(a * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -2e4

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-*.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval N/A

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

      15. sub-neg N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 N/A

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

      18. metadata-eval 8.5

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

   7. Applied rewrites 8.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 98.4%

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

   if -2e4 < (*.f64 a x)

   1. Initial program 25.7%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. flip-- N/A

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

      4. lower-/.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. prod-exp N/A

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

      8. metadata-eval N/A

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

      9. lower-expm1.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(a \cdot x + a \cdot x\right)}}{e^{a \cdot x} + 1} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{a \cdot x} + a \cdot x\right)}{e^{a \cdot x} + 1} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(a \cdot x + \color{blue}{a \cdot x}\right)}{e^{a \cdot x} + 1} \]

      12. distribute-lft-out N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{a \cdot \left(x + x\right)}\right)}{e^{a \cdot x} + 1} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{a \cdot \left(x + x\right)}\right)}{e^{a \cdot x} + 1} \]

      14. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \color{blue}{\left(x + x\right)}\right)}{e^{a \cdot x} + 1} \]

      15. lower-+.f64 99.9

          \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{\color{blue}{e^{a \cdot x} + 1}} \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{e^{a \cdot x} + 1}} \]
   5. Step-by-step derivation

      1. exp-prod N/A

         \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{\color{blue}{{\left(e^{a}\right)}^{x}} + 1} \]

      2. sqr-pow N/A

         \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{\color{blue}{{\left(e^{a}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{a}\right)}^{\left(\frac{x}{2}\right)}} + 1} \]

      3. pow2 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{\color{blue}{{\left({\left(e^{a}\right)}^{\left(\frac{x}{2}\right)}\right)}^{2}} + 1} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{\color{blue}{{\left({\left(e^{a}\right)}^{\left(\frac{x}{2}\right)}\right)}^{2}} + 1} \]

      5. pow-exp N/A

         \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{{\color{blue}{\left(e^{a \cdot \frac{x}{2}}\right)}}^{2} + 1} \]

      6. lower-exp.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{{\color{blue}{\left(e^{a \cdot \frac{x}{2}}\right)}}^{2} + 1} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{{\left(e^{\color{blue}{a \cdot \frac{x}{2}}}\right)}^{2} + 1} \]

      8. div-inv N/A

         \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{{\left(e^{a \cdot \color{blue}{\left(x \cdot \frac{1}{2}\right)}}\right)}^{2} + 1} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{{\left(e^{a \cdot \left(x \cdot \color{blue}{\frac{1}{2}}\right)}\right)}^{2} + 1} \]

      10. lower-*.f64 99.9

          \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{{\left(e^{a \cdot \color{blue}{\left(x \cdot 0.5\right)}}\right)}^{2} + 1} \]

   6. Applied rewrites 99.9%

      \[\leadsto \frac{\mathsf{expm1}\left(a \cdot \left(x + x\right)\right)}{\color{blue}{{\left(e^{a \cdot \left(x \cdot 0.5\right)}\right)}^{2}} + 1} \]

   7. Taylor expanded in a around 0

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. distribute-rgt1-in N/A

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. distribute-lft-in N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. distribute-lft-out N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

   9. Applied rewrites 99.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -20000:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(a, x, -1\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(x, a \cdot \left(x \cdot 0.5\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -20000:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(a, x, -1\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -20000.0) (+ (/ -1.0 (fma a x -1.0)) -1.0) (* a x)))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -20000.0) {
		tmp = (-1.0 / fma(a, x, -1.0)) + -1.0;
	} else {
		tmp = a * x;
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -20000.0)
		tmp = Float64(Float64(-1.0 / fma(a, x, -1.0)) + -1.0);
	else
		tmp = Float64(a * x);
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -20000.0], N[(N[(-1.0 / N[(a * x + -1.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(a * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -2e4

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-*.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval N/A

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

      15. sub-neg N/A

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

      16. lift-*.f64 N/A

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

      17. lower-fma.f64 N/A

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

      18. metadata-eval 8.5

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

   7. Applied rewrites 8.5%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 98.4%

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

   if -2e4 < (*.f64 a x)

   1. Initial program 25.7%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -20000:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(a, x, -1\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.9% accurate, 18.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.6%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 67.2

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

5. Applied rewrites 67.2%

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

Alternative 9: 19.3% accurate, 109.0× speedup

Math

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

FPCore

(FPCore (a x) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(a, x):
	return 0.0

Julia

function code(a, x)
	return 0.0
end

MATLAB

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

Wolfram

code[a_, x_] := 0.0

Derivation

1. Initial program 50.6%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 17.0%

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

   1. metadata-eval 17.0

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

7. Applied rewrites 17.0%

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

Developer Target 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024219 
(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))