expax (section 3.5)

Percentage Accurate: 54.3% → 100.0%

Time: 8.3s

Alternatives: 6

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: 54.3% 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 55.4%

   \[e^{a \cdot x} - 1 \]
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 egg-rr 100.0%

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

Alternative 2: 67.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{+251}:\\ \;\;\;\;a \cdot \left(0.5 \cdot \left(a \cdot \left(x \cdot x\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, a, \left(a \cdot x\right) \cdot \left(a \cdot \left(x \cdot \mathsf{fma}\left(x, a \cdot 0.16666666666666666, 0.5\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= a -1.1e+251)
   (+ (* a (* 0.5 (* a (* x x)))) -1.0)
   (fma x a (* (* a x) (* a (* x (fma x (* a 0.16666666666666666) 0.5)))))))

C

double code(double a, double x) {
	double tmp;
	if (a <= -1.1e+251) {
		tmp = (a * (0.5 * (a * (x * x)))) + -1.0;
	} else {
		tmp = fma(x, a, ((a * x) * (a * (x * fma(x, (a * 0.16666666666666666), 0.5)))));
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (a <= -1.1e+251)
		tmp = Float64(Float64(a * Float64(0.5 * Float64(a * Float64(x * x)))) + -1.0);
	else
		tmp = fma(x, a, Float64(Float64(a * x) * Float64(a * Float64(x * fma(x, Float64(a * 0.16666666666666666), 0.5)))));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[a, -1.1e+251], N[(N[(a * N[(0.5 * N[(a * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(x * a + N[(N[(a * x), $MachinePrecision] * N[(a * N[(x * N[(x * N[(a * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.1e251

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 2.9

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

   5. Simplified 2.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 0.8

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

   7. Applied egg-rr 0.8%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 35.7

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

   10. Simplified 35.7%

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

   if -1.1e251 < a

   1. Initial program 52.2%

      \[e^{a \cdot x} - 1 \]
   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 egg-rr 100.0%

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

   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. 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. lower-fma.f64 N/A

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

   7. Simplified 64.9%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

   9. Applied egg-rr 69.2%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{+251}:\\ \;\;\;\;a \cdot \left(0.5 \cdot \left(a \cdot \left(x \cdot x\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, a, \left(a \cdot x\right) \cdot \left(a \cdot \left(x \cdot \mathsf{fma}\left(x, a \cdot 0.16666666666666666, 0.5\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{+251}:\\ \;\;\;\;a \cdot \left(0.5 \cdot \left(a \cdot \left(x \cdot x\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(\left(a \cdot x\right) \cdot \mathsf{fma}\left(x, a \cdot 0.16666666666666666, 0.5\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= a -1.1e+251)
   (+ (* a (* 0.5 (* a (* x x)))) -1.0)
   (* a (fma (* (* a x) (fma x (* a 0.16666666666666666) 0.5)) x x))))

C

double code(double a, double x) {
	double tmp;
	if (a <= -1.1e+251) {
		tmp = (a * (0.5 * (a * (x * x)))) + -1.0;
	} else {
		tmp = a * fma(((a * x) * fma(x, (a * 0.16666666666666666), 0.5)), x, x);
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (a <= -1.1e+251)
		tmp = Float64(Float64(a * Float64(0.5 * Float64(a * Float64(x * x)))) + -1.0);
	else
		tmp = Float64(a * fma(Float64(Float64(a * x) * fma(x, Float64(a * 0.16666666666666666), 0.5)), x, x));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[a, -1.1e+251], N[(N[(a * N[(0.5 * N[(a * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(a * N[(N[(N[(a * x), $MachinePrecision] * N[(x * N[(a * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.1e251

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 2.9

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

   5. Simplified 2.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 0.8

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

   7. Applied egg-rr 0.8%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 35.7

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

   10. Simplified 35.7%

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

   if -1.1e251 < a

   1. Initial program 52.2%

      \[e^{a \cdot x} - 1 \]
   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 egg-rr 100.0%

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

   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. 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. lower-fma.f64 N/A

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

   7. Simplified 64.9%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 69.2

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 69.2

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

   9. Applied egg-rr 69.2%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{+251}:\\ \;\;\;\;a \cdot \left(0.5 \cdot \left(a \cdot \left(x \cdot x\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(\left(a \cdot x\right) \cdot \mathsf{fma}\left(x, a \cdot 0.16666666666666666, 0.5\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{+250}:\\ \;\;\;\;a \cdot \left(0.5 \cdot \left(a \cdot \left(x \cdot x\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= a -5.8e+250) (+ (* a (* 0.5 (* a (* x x)))) -1.0) (* a x)))

C

double code(double a, double x) {
	double tmp;
	if (a <= -5.8e+250) {
		tmp = (a * (0.5 * (a * (x * x)))) + -1.0;
	} else {
		tmp = a * x;
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if (a <= (-5.8d+250)) then
        tmp = (a * (0.5d0 * (a * (x * x)))) + (-1.0d0)
    else
        tmp = a * x
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if (a <= -5.8e+250) {
		tmp = (a * (0.5 * (a * (x * x)))) + -1.0;
	} else {
		tmp = a * x;
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if a <= -5.8e+250:
		tmp = (a * (0.5 * (a * (x * x)))) + -1.0
	else:
		tmp = a * x
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (a <= -5.8e+250)
		tmp = Float64(Float64(a * Float64(0.5 * Float64(a * Float64(x * x)))) + -1.0);
	else
		tmp = Float64(a * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if (a <= -5.8e+250)
		tmp = (a * (0.5 * (a * (x * x)))) + -1.0;
	else
		tmp = a * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[a, -5.8e+250], N[(N[(a * N[(0.5 * N[(a * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(a * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.80000000000000018e250

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 2.9

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

   5. Simplified 2.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 0.8

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

   7. Applied egg-rr 0.8%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 35.7

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

   10. Simplified 35.7%

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

   if -5.80000000000000018e250 < a

   1. Initial program 52.2%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 68.7

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

   5. Simplified 68.7%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.8 \cdot 10^{+250}:\\ \;\;\;\;a \cdot \left(0.5 \cdot \left(a \cdot \left(x \cdot x\right)\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.7% 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 55.4%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 64.5

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

5. Simplified 64.5%

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

Alternative 6: 19.6% 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 55.4%

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

3. Taylor expanded in a around 0

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

5. Simplified 18.8%

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

   1. metadata-eval 18.8

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

7. Applied egg-rr 18.8%

   \[\leadsto \color{blue}{0} \]
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 2024215 
(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))