expax (section 3.5)

Percentage Accurate: 54.3% → 100.0%

Time: 8.1s

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 56.8%

   \[e^{a \cdot x} - 1 \]
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: 68.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a \cdot x} \leq 0:\\ \;\;\;\;-1 + \frac{\mathsf{fma}\left(a, a \cdot \left(x \cdot x\right), -1\right)}{\mathsf{fma}\left(a, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(a \cdot \left(a \cdot x\right), \mathsf{fma}\left(a, x \cdot 0.16666666666666666, 0.5\right), a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (exp (* a x)) 0.0)
   (+ -1.0 (/ (fma a (* a (* x x)) -1.0) (fma a x -1.0)))
   (* x (fma (* a (* a x)) (fma a (* x 0.16666666666666666) 0.5) a))))

C

double code(double a, double x) {
	double tmp;
	if (exp((a * x)) <= 0.0) {
		tmp = -1.0 + (fma(a, (a * (x * x)), -1.0) / fma(a, x, -1.0));
	} else {
		tmp = x * fma((a * (a * x)), fma(a, (x * 0.16666666666666666), 0.5), a);
	}
	return tmp;
}

Julia

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

Wolfram

code[a_, x_] := If[LessEqual[N[Exp[N[(a * x), $MachinePrecision]], $MachinePrecision], 0.0], N[(-1.0 + N[(N[(a * N[(a * N[(x * x), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(a * x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(a * N[(a * x), $MachinePrecision]), $MachinePrecision] * N[(a * N[(x * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (*.f64 a x)) < 0.0

   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 4.9

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

   5. Simplified 4.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, x, 1\right)} - 1 \]
   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. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. sub-neg N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. metadata-eval 9.4

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

   7. Applied egg-rr 9.4%

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

   if 0.0 < (exp.f64 (*.f64 a x))

   1. Initial program 31.7%

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

      1. lift-*.f64 N/A

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

      2. 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

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

   7. Simplified 99.8%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a \cdot x} \leq 0:\\ \;\;\;\;-1 + \frac{\mathsf{fma}\left(a, a \cdot \left(x \cdot x\right), -1\right)}{\mathsf{fma}\left(a, x, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(a \cdot \left(a \cdot x\right), \mathsf{fma}\left(a, x \cdot 0.16666666666666666, 0.5\right), a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -500000:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{1 + \left(a \cdot x\right) \cdot \mathsf{fma}\left(a, x, -1\right)}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(a \cdot \left(a \cdot x\right), \mathsf{fma}\left(a, x \cdot 0.16666666666666666, 0.5\right), a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -500000.0)
   (fma 1.0 (/ 1.0 (+ 1.0 (* (* a x) (fma a x -1.0)))) -1.0)
   (* x (fma (* a (* a x)) (fma a (* x 0.16666666666666666) 0.5) a))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -500000.0) {
		tmp = fma(1.0, (1.0 / (1.0 + ((a * x) * fma(a, x, -1.0)))), -1.0);
	} else {
		tmp = x * fma((a * (a * x)), fma(a, (x * 0.16666666666666666), 0.5), a);
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -500000.0)
		tmp = fma(1.0, Float64(1.0 / Float64(1.0 + Float64(Float64(a * x) * fma(a, x, -1.0)))), -1.0);
	else
		tmp = Float64(x * fma(Float64(a * Float64(a * x)), fma(a, Float64(x * 0.16666666666666666), 0.5), a));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -500000.0], N[(1.0 * N[(1.0 / N[(1.0 + N[(N[(a * x), $MachinePrecision] * N[(a * x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(x * N[(N[(a * N[(a * x), $MachinePrecision]), $MachinePrecision] * N[(a * N[(x * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -5e5

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 4.9

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

   5. Simplified 4.9%

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

      1. lift-*.f64 4.9

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

      2. +-rgt-identity N/A

         \[\leadsto \color{blue}{a \cdot x + 0} \]

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. flip3-+ N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. div-inv N/A

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

      12. lower-fma.f64 N/A

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

   7. Applied egg-rr 8.0%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 99.8%

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

   if -5e5 < (*.f64 a x)

   1. Initial program 31.7%

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

      1. lift-*.f64 N/A

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

      2. 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

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

   7. Simplified 99.8%

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

Alternative 4: 67.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(a \cdot \left(a \cdot x\right), \mathsf{fma}\left(a, x \cdot 0.16666666666666666, 0.5\right), a\right) \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (* x (fma (* a (* a x)) (fma a (* x 0.16666666666666666) 0.5) a)))

C

double code(double a, double x) {
	return x * fma((a * (a * x)), fma(a, (x * 0.16666666666666666), 0.5), a);
}

Julia

function code(a, x)
	return Float64(x * fma(Float64(a * Float64(a * x)), fma(a, Float64(x * 0.16666666666666666), 0.5), a))
end

Wolfram

code[a_, x_] := N[(x * N[(N[(a * N[(a * x), $MachinePrecision]), $MachinePrecision] * N[(a * N[(x * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.8%

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

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

7. Simplified 64.5%

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

Alternative 5: 66.8% 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 56.8%

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

3. Taylor expanded in a around 0

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

   1. lower-*.f64 63.9

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

5. Simplified 63.9%

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

Alternative 6: 19.7% 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 56.8%

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

3. Taylor expanded in a around 0

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

5. Simplified 19.4%

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

   1. metadata-eval 19.4

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

7. Applied egg-rr 19.4%

   \[\leadsto \color{blue}{0} \]
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 2024207 
(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))