expax (section 3.5)

Percentage Accurate: 54.1% → 100.0%

Time: 8.6s

Alternatives: 9

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: 54.1% 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 53.1%

   \[e^{a \cdot x} - 1 \]
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: 99.2% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -5.0)
   -1.0
   (* a (fma (* (* a x) (fma a (* x 0.16666666666666666) 0.5)) x x))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -5.0) {
		tmp = -1.0;
	} else {
		tmp = a * fma(((a * x) * fma(a, (x * 0.16666666666666666), 0.5)), x, x);
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -5.0)
		tmp = -1.0;
	else
		tmp = Float64(a * fma(Float64(Float64(a * x) * fma(a, Float64(x * 0.16666666666666666), 0.5)), x, x));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -5.0], -1.0, N[(a * N[(N[(N[(a * x), $MachinePrecision] * N[(a * N[(x * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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. accelerator-lowering-fma.f64 5.3

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

   5. Simplified 5.3%

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

      1. flip-+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

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

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

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

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

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

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

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

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

      14. metadata-eval 9.1

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

   7. Applied egg-rr 9.1%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 96.1%

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

   11. Taylor expanded in a around inf

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

   13. Simplified 98.8%

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

   if -5 < (*.f64 a x)

   1. Initial program 28.6%

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

   3. Taylor expanded in a around 0

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

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

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. associate-*r* N/A

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

      10. +-commutative N/A

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

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

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

   5. Simplified 93.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 99.8

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

   7. Applied egg-rr 99.8%

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

4. Final simplification 99.5%

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

Alternative 3: 98.9% accurate, 2.9× speedup

Math

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

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -5.0) -1.0 (fma x a (* a (* 0.5 (* x (* a x)))))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -5.0) {
		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) <= -5.0)
		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], -5.0], -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) < -5

   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. accelerator-lowering-fma.f64 5.3

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

   5. Simplified 5.3%

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

      1. flip-+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

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

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

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

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

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

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

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

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

      14. metadata-eval 9.1

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

   7. Applied egg-rr 9.1%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 96.1%

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

   11. Taylor expanded in a around inf

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

   13. Simplified 98.8%

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

   if -5 < (*.f64 a x)

   1. Initial program 28.6%

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

   3. Taylor expanded in a around 0

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

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

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. associate-*r* N/A

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

      10. +-commutative N/A

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

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

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

   5. Simplified 93.0%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 92.9%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-lowering-*.f64 99.7

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

   10. Applied egg-rr 99.7%

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

4. Final simplification 99.4%

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

Alternative 4: 98.9% accurate, 3.3× speedup

Math

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

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -5.0) -1.0 (* x (fma (* a (* x 0.5)) a a))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -5.0) {
		tmp = -1.0;
	} else {
		tmp = x * fma((a * (x * 0.5)), a, a);
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -5.0)
		tmp = -1.0;
	else
		tmp = Float64(x * fma(Float64(a * Float64(x * 0.5)), a, a));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -5.0], -1.0, N[(x * N[(N[(a * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] * a + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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. accelerator-lowering-fma.f64 5.3

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

   5. Simplified 5.3%

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

      1. flip-+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

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

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

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

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

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

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

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

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

      14. metadata-eval 9.1

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

   7. Applied egg-rr 9.1%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 96.1%

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

   11. Taylor expanded in a around inf

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

   13. Simplified 98.8%

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

   if -5 < (*.f64 a x)

   1. Initial program 28.6%

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

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{expm1}\left(\color{blue}{a \cdot x + 0}\right) \]

      2. metadata-eval N/A

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

      3. associate--l+ N/A

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

      4. flip-- N/A

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

      5. clear-num N/A

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

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

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. associate--l+ N/A

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

      10. metadata-eval N/A

          \[\leadsto \mathsf{expm1}\left(\frac{1}{\frac{1}{a \cdot x + \color{blue}{0}}}\right) \]

      11. +-rgt-identity N/A

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

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

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

      13. *-lowering-*.f64 99.0

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in a around 0

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. associate-*l* N/A

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

      11. distribute-rgt-in N/A

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

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

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

      13. +-commutative N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

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

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

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

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

      18. unpow2 N/A

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

      19. *-lowering-*.f64 93.1

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

   9. Simplified 93.1%

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

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

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

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

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

       5. *-lowering-*.f64 99.7

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

   11. Applied egg-rr 99.7%

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

4. Final simplification 99.4%

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

Alternative 5: 98.9% accurate, 3.3× speedup

Math

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

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -5.0) -1.0 (* a (fma (* x (* a x)) 0.5 x))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -5.0) {
		tmp = -1.0;
	} else {
		tmp = a * fma((x * (a * x)), 0.5, x);
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -5.0)
		tmp = -1.0;
	else
		tmp = Float64(a * fma(Float64(x * Float64(a * x)), 0.5, x));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -5.0], -1.0, N[(a * N[(N[(x * N[(a * x), $MachinePrecision]), $MachinePrecision] * 0.5 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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. accelerator-lowering-fma.f64 5.3

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

   5. Simplified 5.3%

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

      1. flip-+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

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

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

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

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

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

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

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

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

      14. metadata-eval 9.1

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

   7. Applied egg-rr 9.1%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 96.1%

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

   11. Taylor expanded in a around inf

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

   13. Simplified 98.8%

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

   if -5 < (*.f64 a x)

   1. Initial program 28.6%

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

   3. Taylor expanded in a around 0

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

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

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. associate-*r* N/A

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

      10. +-commutative N/A

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

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

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

   5. Simplified 93.0%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 92.9%

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 99.7

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

   10. Applied egg-rr 99.7%

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

4. Final simplification 99.4%

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

Alternative 6: 98.2% accurate, 3.9× speedup

Math

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

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -5.0) -1.0 (* x (fma a (* a x) a))))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -5.0) {
		tmp = -1.0;
	} else {
		tmp = x * fma(a, (a * x), a);
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -5.0)
		tmp = -1.0;
	else
		tmp = Float64(x * fma(a, Float64(a * x), a));
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -5.0], -1.0, N[(x * N[(a * N[(a * x), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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. accelerator-lowering-fma.f64 5.3

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

   5. Simplified 5.3%

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

      1. flip-+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

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

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

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

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

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

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

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

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

      14. metadata-eval 9.1

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

   7. Applied egg-rr 9.1%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 96.1%

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

   11. Taylor expanded in a around inf

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

   13. Simplified 98.8%

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

   if -5 < (*.f64 a x)

   1. Initial program 28.6%

      \[e^{a \cdot x} - 1 \]
   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. accelerator-lowering-fma.f64 28.3

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

   5. Simplified 28.3%

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

      1. flip-+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

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

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

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

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

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

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

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

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

      14. metadata-eval 27.7

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

   7. Applied egg-rr 27.7%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 28.2%

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

   11. Taylor expanded in a around 0

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

       1. distribute-lft-in N/A

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

       2. associate-*r* N/A

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

       3. unpow2 N/A

          \[\leadsto a \cdot x + \color{blue}{{a}^{2}} \cdot {x}^{2} \]

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. distribute-rgt-in N/A

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

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

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

       8. +-commutative N/A

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

       9. unpow2 N/A

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

       10. associate-*l* N/A

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

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

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

       12. *-lowering-*.f64 98.6

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

   13. Simplified 98.6%

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

Alternative 7: 98.2% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (a x) :precision binary64 (if (<= (* a x) -5.0) -1.0 (* a x)))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -5.0) {
		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) <= (-5.0d0)) 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) <= -5.0) {
		tmp = -1.0;
	} else {
		tmp = a * x;
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (a * x) <= -5.0:
		tmp = -1.0
	else:
		tmp = a * x
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -5.0)
		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) <= -5.0)
		tmp = -1.0;
	else
		tmp = a * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -5.0], -1.0, N[(a * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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. accelerator-lowering-fma.f64 5.3

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

   5. Simplified 5.3%

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

      1. flip-+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

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

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

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

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

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

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

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

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

      14. metadata-eval 9.1

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

   7. Applied egg-rr 9.1%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 96.1%

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

   11. Taylor expanded in a around inf

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

   13. Simplified 98.8%

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

   if -5 < (*.f64 a x)

   1. Initial program 28.6%

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

   3. Taylor expanded in a around 0

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

      1. *-lowering-*.f64 98.6

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

   5. Simplified 98.6%

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

Alternative 8: 52.1% 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 74.4%

      \[e^{a \cdot x} - 1 \]
   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. accelerator-lowering-fma.f64 6.2

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

   5. Simplified 6.2%

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

      1. flip-+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. sub-neg N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

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

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

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

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

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

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

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

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

      14. metadata-eval 8.6

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

   7. Applied egg-rr 8.6%

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

   8. Taylor expanded in a around 0

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

   10. Simplified 71.1%

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

   11. Taylor expanded in a around inf

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

   13. Simplified 72.7%

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

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

   1. Initial program 33.4%

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

   3. Taylor expanded in a around 0

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

   5. Simplified 32.3%

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

      1. metadata-eval 32.3

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

   7. Applied egg-rr 32.3%

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

Alternative 9: 36.3% 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 53.1%

   \[e^{a \cdot x} - 1 \]
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. accelerator-lowering-fma.f64 20.4

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

5. Simplified 20.4%

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

   1. flip-+ N/A

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

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

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

   3. metadata-eval N/A

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

   4. sub-neg N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

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

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

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

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

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

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

   11. metadata-eval N/A

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

   12. sub-neg N/A

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

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

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

   14. metadata-eval 21.3

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

7. Applied egg-rr 21.3%

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

8. Taylor expanded in a around 0

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

10. Simplified 51.5%

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

11. Taylor expanded in a around inf

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

13. Simplified 36.4%

    \[\leadsto \color{blue}{-1} \]
14. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024198 
(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))