expax (section 3.5)

Percentage Accurate: 53.4% → 100.0%

Time: 9.0s

Alternatives: 8

Speedup: 6.4×

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: 53.4% 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 48.8%

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

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

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

   2. *-lowering-*.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 99.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -10:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(\left(a \cdot x\right) \cdot \mathsf{fma}\left(a \cdot x, \mathsf{fma}\left(x, a \cdot 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -10.0)
   -1.0
   (*
    a
    (fma
     (*
      (* a x)
      (fma (* a x) (fma x (* a 0.041666666666666664) 0.16666666666666666) 0.5))
     x
     x))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -10.0) {
		tmp = -1.0;
	} else {
		tmp = a * fma(((a * x) * fma((a * x), fma(x, (a * 0.041666666666666664), 0.16666666666666666), 0.5)), x, x);
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -10.0)
		tmp = -1.0;
	else
		tmp = Float64(a * fma(Float64(Float64(a * x) * fma(Float64(a * x), fma(x, Float64(a * 0.041666666666666664), 0.16666666666666666), 0.5)), x, x));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -10.0], -1.0, N[(a * N[(N[(N[(a * x), $MachinePrecision] * N[(N[(a * x), $MachinePrecision] * N[(x * N[(a * 0.041666666666666664), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -10

   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. accelerator-lowering-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 1.5

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

   5. Simplified 1.5%

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

   6. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 10.2

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

   8. Simplified 10.2%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 99.3%

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

   if -10 < (*.f64 a x)

   1. Initial program 27.9%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 92.0%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

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

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

      14. *-lowering-*.f64 100.0

          \[\leadsto a \cdot \mathsf{fma}\left(\left(x \cdot a\right) \cdot \mathsf{fma}\left(x \cdot a, \mathsf{fma}\left(x, \color{blue}{a \cdot 0.041666666666666664}, 0.16666666666666666\right), 0.5\right), x, x\right) \]

   7. Applied egg-rr 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -10:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(\left(a \cdot x\right) \cdot \mathsf{fma}\left(a \cdot x, \mathsf{fma}\left(x, a \cdot 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -10.0)
   -1.0
   (* a (fma (* a x) (* x (fma (* a x) 0.16666666666666666 0.5)) x))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -10.0) {
		tmp = -1.0;
	} else {
		tmp = a * fma((a * x), (x * fma((a * x), 0.16666666666666666, 0.5)), x);
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -10.0)
		tmp = -1.0;
	else
		tmp = Float64(a * fma(Float64(a * x), Float64(x * fma(Float64(a * x), 0.16666666666666666, 0.5)), x));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -10.0], -1.0, N[(a * N[(N[(a * x), $MachinePrecision] * N[(x * N[(N[(a * x), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -10

   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. accelerator-lowering-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 1.5

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

   5. Simplified 1.5%

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

   6. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 10.2

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

   8. Simplified 10.2%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 99.3%

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

   if -10 < (*.f64 a x)

   1. Initial program 27.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 26.6%

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

      1. associate--l+ N/A

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

      2. metadata-eval N/A

         \[\leadsto a \cdot \left(\left(a \cdot \left(x \cdot x\right)\right) \cdot \left(a \cdot \left(x \cdot \frac{1}{6}\right) + \frac{1}{2}\right) + x\right) + \color{blue}{0} \]

      3. +-rgt-identity N/A

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

      4. *-commutative N/A

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

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

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

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

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

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

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

      11. associate-*r* N/A

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

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

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

      13. *-lowering-*.f64 99.8

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

   7. Applied egg-rr 99.8%

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

4. Final simplification 99.6%

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

Alternative 4: 99.1% accurate, 2.9× speedup

Math

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

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -10.0) -1.0 (fma x a (* a (* 0.5 (* x (* a x)))))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -10.0) {
		tmp = -1.0;
	} else {
		tmp = fma(x, a, (a * (0.5 * (x * (a * x)))));
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -10.0)
		tmp = -1.0;
	else
		tmp = fma(x, a, Float64(a * Float64(0.5 * Float64(x * Float64(a * x)))));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -10.0], -1.0, N[(x * a + N[(a * N[(0.5 * N[(x * N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -10

   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. accelerator-lowering-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 1.5

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

   5. Simplified 1.5%

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

   6. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 10.2

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

   8. Simplified 10.2%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 99.3%

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

   if -10 < (*.f64 a x)

   1. Initial program 27.9%

      \[e^{a \cdot x} - 1 \]
   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. accelerator-lowering-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 26.5

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

   5. Simplified 26.5%

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

      1. associate--l+ N/A

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

      2. metadata-eval N/A

         \[\leadsto a \cdot \left(a \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right) + x\right) + \color{blue}{0} \]

      3. +-rgt-identity N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

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

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

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

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

      16. *-lowering-*.f64 99.4

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

   7. Applied egg-rr 99.4%

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

Alternative 5: 99.1% accurate, 3.3× speedup

Math

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

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -10.0) -1.0 (* a (fma 0.5 (* x (* a x)) x))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -10.0) {
		tmp = -1.0;
	} else {
		tmp = a * fma(0.5, (x * (a * x)), x);
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -10.0)
		tmp = -1.0;
	else
		tmp = Float64(a * fma(0.5, Float64(x * Float64(a * x)), x));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -10.0], -1.0, N[(a * N[(0.5 * N[(x * N[(a * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -10

   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. accelerator-lowering-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 1.5

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

   5. Simplified 1.5%

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

   6. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 10.2

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

   8. Simplified 10.2%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 99.3%

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

   if -10 < (*.f64 a x)

   1. Initial program 27.9%

      \[e^{a \cdot x} - 1 \]
   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. accelerator-lowering-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 26.5

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

   5. Simplified 26.5%

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

      1. associate--l+ N/A

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

      2. metadata-eval N/A

         \[\leadsto a \cdot \left(a \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right) + x\right) + \color{blue}{0} \]

      3. +-rgt-identity N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

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

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

      13. *-lowering-*.f64 99.4

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

   7. Applied egg-rr 99.4%

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

4. Final simplification 99.3%

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

Alternative 6: 98.4% accurate, 6.4× speedup

Math

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

FPCore

(FPCore (a x) :precision binary64 (if (<= (* a x) -10.0) -1.0 (* a x)))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -10.0) {
		tmp = -1.0;
	} else {
		tmp = a * x;
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((a * x) <= (-10.0d0)) then
        tmp = -1.0d0
    else
        tmp = a * x
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -10.0) {
		tmp = -1.0;
	} else {
		tmp = a * x;
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -10.0:
		tmp = -1.0
	else:
		tmp = a * x
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -10.0)
		tmp = -1.0;
	else
		tmp = Float64(a * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -10.0)
		tmp = -1.0;
	else
		tmp = a * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -10.0], -1.0, N[(a * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -10

   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. accelerator-lowering-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 1.5

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

   5. Simplified 1.5%

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

   6. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 10.2

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

   8. Simplified 10.2%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 99.3%

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

   if -10 < (*.f64 a x)

   1. Initial program 27.9%

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

   3. Taylor expanded in a around 0

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

      1. *-lowering-*.f64 97.9

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

   5. Simplified 97.9%

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

Alternative 7: 51.7% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -1.1 \cdot 10^{-154}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a x) :precision binary64 (if (<= (* a x) -1.1e-154) -1.0 0.0))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -1.1e-154) {
		tmp = -1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, x)
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((a * x) <= (-1.1d-154)) then
        tmp = -1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((a * x) <= -1.1e-154) {
		tmp = -1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -1.1e-154:
		tmp = -1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -1.1e-154)
		tmp = -1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((a * x) <= -1.1e-154)
		tmp = -1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -1.1e-154], -1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -1.10000000000000004e-154

   1. Initial program 63.1%

      \[e^{a \cdot x} - 1 \]
   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. accelerator-lowering-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 3.9

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

   5. Simplified 3.9%

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

   6. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 8.3

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

   8. Simplified 8.3%

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

   9. Taylor expanded in x around 0

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

   11. Simplified 61.6%

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

   if -1.10000000000000004e-154 < (*.f64 a x)

   1. Initial program 35.1%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 32.7%

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

      1. metadata-eval 32.7

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

   7. Applied egg-rr 32.7%

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

Alternative 8: 35.2% accurate, 109.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, x):
	return -1.0

Julia

function code(a, x)
	return -1.0
end

MATLAB

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

Wolfram

code[a_, x_] := -1.0

Derivation

1. Initial program 48.8%

   \[e^{a \cdot x} - 1 \]
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. accelerator-lowering-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 19.3

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

5. Simplified 19.3%

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

6. Taylor expanded in a around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

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

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

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

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 5.4

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

8. Simplified 5.4%

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

9. Taylor expanded in x around 0

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

11. Simplified 31.4%

    \[\leadsto \color{blue}{-1} \]
12. 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 2024199 
(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))