Kahan's exp quotient

Percentage Accurate: 52.9% → 100.0%

Time: 11.0s

Alternatives: 18

Speedup: 8.8×

• 24.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{e^{x} - 1}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (exp x) 1.0) x))

C

double code(double x) {
	return (exp(x) - 1.0) / x;
}

Fortran

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

Java

public static double code(double x) {
	return (Math.exp(x) - 1.0) / x;
}

Python

def code(x):
	return (math.exp(x) - 1.0) / x

Julia

function code(x)
	return Float64(Float64(exp(x) - 1.0) / x)
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - 1.0) / x;
end

Wolfram

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision] / x), $MachinePrecision]

Initial Program: 52.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x} - 1}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (exp x) 1.0) x))

C

double code(double x) {
	return (exp(x) - 1.0) / x;
}

Fortran

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

Java

public static double code(double x) {
	return (Math.exp(x) - 1.0) / x;
}

Python

def code(x):
	return (math.exp(x) - 1.0) / x

Julia

function code(x)
	return Float64(Float64(exp(x) - 1.0) / x)
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - 1.0) / x;
end

Wolfram

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision] / x), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (expm1 x) x))

C

double code(double x) {
	return expm1(x) / x;
}

Java

public static double code(double x) {
	return Math.expm1(x) / x;
}

Python

def code(x):
	return math.expm1(x) / x

Julia

function code(x)
	return Float64(expm1(x) / x)
end

Wolfram

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

Derivation

1. Initial program 53.3%

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

   1. accelerator-lowering-expm1.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 70.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right)\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;x \leq 1.65 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{x \cdot \mathsf{fma}\left(t\_1, t\_1, -1\right)}{\mathsf{fma}\left(x, t\_0, -1\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x (fma x 0.041666666666666664 0.16666666666666666) 0.5))
        (t_1 (* x t_0)))
   (if (<= x 1.65e+103)
     (/ (/ (* x (fma t_1 t_1 -1.0)) (fma x t_0 -1.0)) x)
     (* x (* x (* x 0.041666666666666664))))))

C

double code(double x) {
	double t_0 = fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5);
	double t_1 = x * t_0;
	double tmp;
	if (x <= 1.65e+103) {
		tmp = ((x * fma(t_1, t_1, -1.0)) / fma(x, t_0, -1.0)) / x;
	} else {
		tmp = x * (x * (x * 0.041666666666666664));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5)
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if (x <= 1.65e+103)
		tmp = Float64(Float64(Float64(x * fma(t_1, t_1, -1.0)) / fma(x, t_0, -1.0)) / x);
	else
		tmp = Float64(x * Float64(x * Float64(x * 0.041666666666666664)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, If[LessEqual[x, 1.65e+103], N[(N[(N[(x * N[(t$95$1 * t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision] / N[(x * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.65000000000000004e103

   1. Initial program 43.9%

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

      1. accelerator-lowering-expm1.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}}{x} \]
   6. Step-by-step derivation

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

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

      2. +-commutative N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 63.6

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)}, 0.5\right), 1\right)}{x} \]

   7. Simplified 63.6%

      \[\leadsto \frac{\color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right)}}{x} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   9. Applied egg-rr 66.9%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), -1\right) \cdot x}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), -1\right)}}}{x} \]

   if 1.65000000000000004e103 < x

   1. Initial program 100.0%

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

      1. accelerator-lowering-expm1.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}}{x} \]
   6. Step-by-step derivation

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

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

      2. +-commutative N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 100.0

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)}, 0.5\right), 1\right)}{x} \]

   7. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right)}}{x} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{3}} \]
   9. Step-by-step derivation

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 100.0

          \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(x \cdot 0.041666666666666664\right)}\right) \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{x \cdot \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), -1\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), -1\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)\\ \mathbf{if}\;x \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), \left(\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right) \cdot \left(x \cdot x\right)\right) \cdot t\_0, 1\right)}{\mathsf{fma}\left(t\_0, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), -1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (fma x 0.16666666666666666 0.5))))
   (if (<= x 2e+77)
     (/
      (fma
       (fma x 0.16666666666666666 0.5)
       (* (* (fma x 0.16666666666666666 0.5) (* x x)) t_0)
       1.0)
      (fma t_0 (fma x (fma x 0.16666666666666666 0.5) -1.0) 1.0))
     (/ (* 0.041666666666666664 (* x (* x (* x x)))) x))))

C

double code(double x) {
	double t_0 = x * fma(x, 0.16666666666666666, 0.5);
	double tmp;
	if (x <= 2e+77) {
		tmp = fma(fma(x, 0.16666666666666666, 0.5), ((fma(x, 0.16666666666666666, 0.5) * (x * x)) * t_0), 1.0) / fma(t_0, fma(x, fma(x, 0.16666666666666666, 0.5), -1.0), 1.0);
	} else {
		tmp = (0.041666666666666664 * (x * (x * (x * x)))) / x;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * fma(x, 0.16666666666666666, 0.5))
	tmp = 0.0
	if (x <= 2e+77)
		tmp = Float64(fma(fma(x, 0.16666666666666666, 0.5), Float64(Float64(fma(x, 0.16666666666666666, 0.5) * Float64(x * x)) * t_0), 1.0) / fma(t_0, fma(x, fma(x, 0.16666666666666666, 0.5), -1.0), 1.0));
	else
		tmp = Float64(Float64(0.041666666666666664 * Float64(x * Float64(x * Float64(x * x)))) / x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2e+77], N[(N[(N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(N[(N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.041666666666666664 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999999999997e77

   1. Initial program 41.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 62.6

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right) \]

   5. Simplified 62.6%

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

      1. flip3-+ N/A

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

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

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

   7. Applied egg-rr 65.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), \left(\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)\right), 1\right)}{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), -1\right), 1\right)}} \]

   if 1.99999999999999997e77 < x

   1. Initial program 100.0%

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

      1. accelerator-lowering-expm1.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}}{x} \]
   6. Step-by-step derivation

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

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

      2. +-commutative N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 100.0

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)}, 0.5\right), 1\right)}{x} \]

   7. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right)}}{x} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{24} \cdot {x}^{4}}}{x} \]
   9. Step-by-step derivation

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

         \[\leadsto \frac{\color{blue}{\frac{1}{24} \cdot {x}^{4}}}{x} \]

      2. metadata-eval N/A

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

      3. pow-plus N/A

         \[\leadsto \frac{\frac{1}{24} \cdot \color{blue}{\left({x}^{3} \cdot x\right)}}{x} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{1}{24} \cdot \color{blue}{\left(x \cdot {x}^{3}\right)}}{x} \]

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

         \[\leadsto \frac{\frac{1}{24} \cdot \color{blue}{\left(x \cdot {x}^{3}\right)}}{x} \]

      6. cube-mult N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 100.0

          \[\leadsto \frac{0.041666666666666664 \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)}{x} \]

   10. Simplified 100.0%

       \[\leadsto \frac{\color{blue}{0.041666666666666664 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 70.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right)\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;x \leq 1.65 \cdot 10^{+103}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, t\_1, -1\right)}{\mathsf{fma}\left(x, t\_0, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x (fma x 0.041666666666666664 0.16666666666666666) 0.5))
        (t_1 (* x t_0)))
   (if (<= x 1.65e+103)
     (/ (fma t_1 t_1 -1.0) (fma x t_0 -1.0))
     (* x (* x (* x 0.041666666666666664))))))

C

double code(double x) {
	double t_0 = fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5);
	double t_1 = x * t_0;
	double tmp;
	if (x <= 1.65e+103) {
		tmp = fma(t_1, t_1, -1.0) / fma(x, t_0, -1.0);
	} else {
		tmp = x * (x * (x * 0.041666666666666664));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5)
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if (x <= 1.65e+103)
		tmp = Float64(fma(t_1, t_1, -1.0) / fma(x, t_0, -1.0));
	else
		tmp = Float64(x * Float64(x * Float64(x * 0.041666666666666664)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, If[LessEqual[x, 1.65e+103], N[(N[(t$95$1 * t$95$1 + -1.0), $MachinePrecision] / N[(x * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.65000000000000004e103

   1. Initial program 43.9%

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

      1. accelerator-lowering-expm1.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}}{x} \]
   6. Step-by-step derivation

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

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

      2. +-commutative N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 63.6

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)}, 0.5\right), 1\right)}{x} \]

   7. Simplified 63.6%

      \[\leadsto \frac{\color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right)}}{x} \]
   8. Step-by-step derivation

      1. flip-+ N/A

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

      2. div-inv N/A

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

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

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

      4. sub-neg N/A

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

      5. swap-sqr N/A

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

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

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

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

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

      8. metadata-eval N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{576}}, \mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{6}\right)\right) \cdot \frac{1}{x \cdot \frac{1}{24} - \frac{1}{6}}, \frac{1}{2}\right), 1\right)}{x} \]

      9. metadata-eval N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{1}{576}, \mathsf{neg}\left(\color{blue}{\frac{1}{36}}\right)\right) \cdot \frac{1}{x \cdot \frac{1}{24} - \frac{1}{6}}, \frac{1}{2}\right), 1\right)}{x} \]

      10. metadata-eval N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{1}{576}, \color{blue}{\frac{-1}{36}}\right) \cdot \frac{1}{x \cdot \frac{1}{24} - \frac{1}{6}}, \frac{1}{2}\right), 1\right)}{x} \]

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

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{1}{576}, \frac{-1}{36}\right) \cdot \color{blue}{\frac{1}{x \cdot \frac{1}{24} - \frac{1}{6}}}, \frac{1}{2}\right), 1\right)}{x} \]

      12. sub-neg N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{1}{576}, \frac{-1}{36}\right) \cdot \frac{1}{\color{blue}{x \cdot \frac{1}{24} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}}, \frac{1}{2}\right), 1\right)}{x} \]

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

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \frac{1}{576}, \frac{-1}{36}\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(x, \frac{1}{24}, \mathsf{neg}\left(\frac{1}{6}\right)\right)}}, \frac{1}{2}\right), 1\right)}{x} \]

      14. metadata-eval 63.6

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, 0.001736111111111111, -0.027777777777777776\right) \cdot \frac{1}{\mathsf{fma}\left(x, 0.041666666666666664, \color{blue}{-0.16666666666666666}\right)}, 0.5\right), 1\right)}{x} \]

   9. Applied egg-rr 63.6%

      \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x \cdot x, 0.001736111111111111, -0.027777777777777776\right) \cdot \frac{1}{\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right)}}, 0.5\right), 1\right)}{x} \]

   11. Applied egg-rr 66.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), -1\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), -1\right)}} \]

   if 1.65000000000000004e103 < x

   1. Initial program 100.0%

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

      1. accelerator-lowering-expm1.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}}{x} \]
   6. Step-by-step derivation

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

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

      2. +-commutative N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 100.0

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)}, 0.5\right), 1\right)}{x} \]

   7. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right)}}{x} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{3}} \]
   9. Step-by-step derivation

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 100.0

          \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(x \cdot 0.041666666666666664\right)}\right) \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 69.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.9:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.9) 1.0 (/ (* x (* x (* x (* x 0.041666666666666664)))) x)))

C

double code(double x) {
	double tmp;
	if (x <= 2.9) {
		tmp = 1.0;
	} else {
		tmp = (x * (x * (x * (x * 0.041666666666666664)))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.9d0) then
        tmp = 1.0d0
    else
        tmp = (x * (x * (x * (x * 0.041666666666666664d0)))) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.9) {
		tmp = 1.0;
	} else {
		tmp = (x * (x * (x * (x * 0.041666666666666664)))) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.9:
		tmp = 1.0
	else:
		tmp = (x * (x * (x * (x * 0.041666666666666664)))) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.9)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x * Float64(x * Float64(x * Float64(x * 0.041666666666666664)))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.9)
		tmp = 1.0;
	else
		tmp = (x * (x * (x * (x * 0.041666666666666664)))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.9], 1.0, N[(N[(x * N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.89999999999999991

   1. Initial program 35.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 68.6%

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

   if 2.89999999999999991 < x

   1. Initial program 100.0%

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

      1. accelerator-lowering-expm1.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}}{x} \]
   6. Step-by-step derivation

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

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

      2. +-commutative N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 74.1

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)}, 0.5\right), 1\right)}{x} \]

   7. Simplified 74.1%

      \[\leadsto \frac{\color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right)}}{x} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)}}{x} \]
   9. Step-by-step derivation

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 74.1

          \[\leadsto \frac{x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot 0.041666666666666664\right)}\right)\right)}{x} \]

   10. Simplified 74.1%

       \[\leadsto \frac{x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right)\right)}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 69.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.9:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.041666666666666664 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.9) 1.0 (/ (* 0.041666666666666664 (* x (* x (* x x)))) x)))

C

double code(double x) {
	double tmp;
	if (x <= 2.9) {
		tmp = 1.0;
	} else {
		tmp = (0.041666666666666664 * (x * (x * (x * x)))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.9d0) then
        tmp = 1.0d0
    else
        tmp = (0.041666666666666664d0 * (x * (x * (x * x)))) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.9) {
		tmp = 1.0;
	} else {
		tmp = (0.041666666666666664 * (x * (x * (x * x)))) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.9:
		tmp = 1.0
	else:
		tmp = (0.041666666666666664 * (x * (x * (x * x)))) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.9)
		tmp = 1.0;
	else
		tmp = Float64(Float64(0.041666666666666664 * Float64(x * Float64(x * Float64(x * x)))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.9)
		tmp = 1.0;
	else
		tmp = (0.041666666666666664 * (x * (x * (x * x)))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.9], 1.0, N[(N[(0.041666666666666664 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.89999999999999991

   1. Initial program 35.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 68.6%

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

   if 2.89999999999999991 < x

   1. Initial program 100.0%

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

      1. accelerator-lowering-expm1.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}}{x} \]
   6. Step-by-step derivation

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

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

      2. +-commutative N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 74.1

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)}, 0.5\right), 1\right)}{x} \]

   7. Simplified 74.1%

      \[\leadsto \frac{\color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right)}}{x} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{24} \cdot {x}^{4}}}{x} \]
   9. Step-by-step derivation

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

         \[\leadsto \frac{\color{blue}{\frac{1}{24} \cdot {x}^{4}}}{x} \]

      2. metadata-eval N/A

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

      3. pow-plus N/A

         \[\leadsto \frac{\frac{1}{24} \cdot \color{blue}{\left({x}^{3} \cdot x\right)}}{x} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{1}{24} \cdot \color{blue}{\left(x \cdot {x}^{3}\right)}}{x} \]

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

         \[\leadsto \frac{\frac{1}{24} \cdot \color{blue}{\left(x \cdot {x}^{3}\right)}}{x} \]

      6. cube-mult N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 74.1

          \[\leadsto \frac{0.041666666666666664 \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)}{x} \]

   10. Simplified 74.1%

       \[\leadsto \frac{\color{blue}{0.041666666666666664 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 68.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right)}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (*
   x
   (fma x (fma x (fma x 0.041666666666666664 0.16666666666666666) 0.5) 1.0))
  x))

C

double code(double x) {
	return (x * fma(x, fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5), 1.0)) / x;
}

Julia

function code(x)
	return Float64(Float64(x * fma(x, fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5), 1.0)) / x)
end

Wolfram

code[x_] := N[(N[(x * N[(x * N[(x * N[(x * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 53.3%

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

   1. accelerator-lowering-expm1.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}}{x} \]
6. Step-by-step derivation

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

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

   2. +-commutative N/A

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

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

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

   4. +-commutative N/A

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

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

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f64 69.7

      \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)}, 0.5\right), 1\right)}{x} \]

7. Simplified 69.7%

   \[\leadsto \frac{\color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right)}}{x} \]
8. Add Preprocessing

Alternative 8: 66.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.8:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.125, 0.25\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.8) 1.0 (* x (fma x (fma x 0.125 0.25) 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= 0.8) {
		tmp = 1.0;
	} else {
		tmp = x * fma(x, fma(x, 0.125, 0.25), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.8)
		tmp = 1.0;
	else
		tmp = Float64(x * fma(x, fma(x, 0.125, 0.25), 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.8], 1.0, N[(x * N[(x * N[(x * 0.125 + 0.25), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.80000000000000004

   1. Initial program 35.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 68.6%

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

   if 0.80000000000000004 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot x + 1} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{2}} + 1 \]

      3. accelerator-lowering-fma.f64 6.0

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

   5. Simplified 6.0%

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

      1. flip-+ N/A

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

      2. metadata-eval N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. flip3-- N/A

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

      6. associate-/r/ N/A

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

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

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

   7. Applied egg-rr 1.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x \cdot \left(x \cdot 0.25\right)}{\mathsf{fma}\left(0.125, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.25, 0.5\right), 1\right), -\frac{1}{\mathsf{fma}\left(x, 0.5, -1\right)}\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \color{blue}{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot x\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot x\right)\right) + 1}\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot x\right), 1\right)}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot x\right) + \frac{1}{2}}, 1\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{4} + \frac{1}{8} \cdot x, \frac{1}{2}\right)}, 1\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{8} \cdot x + \frac{1}{4}}, \frac{1}{2}\right), 1\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{8}} + \frac{1}{4}, \frac{1}{2}\right), 1\right)\right) \]

      7. accelerator-lowering-fma.f64 8.0

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot 0.25\right)}{\mathsf{fma}\left(0.125, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.25, 0.5\right), 1\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.125, 0.25\right)}, 0.5\right), 1\right)\right) \]

   10. Simplified 8.0%

       \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot 0.25\right)}{\mathsf{fma}\left(0.125, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.25, 0.5\right), 1\right), \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.125, 0.25\right), 0.5\right), 1\right)}\right) \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{1}{8} + \left(\frac{1}{4} \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)} \]
   12. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(\frac{1}{8} + \left(\frac{1}{4} \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\frac{1}{8} + \left(\frac{1}{4} \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{8} + \left(\frac{1}{4} \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)\right)} \]

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

          \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{8} + \left(\frac{1}{4} \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)\right)} \]

       5. associate-+r+ N/A

          \[\leadsto x \cdot \left({x}^{2} \cdot \color{blue}{\left(\left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right) + \frac{1}{{x}^{2}}\right)}\right) \]

       6. distribute-lft-in N/A

          \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right) + {x}^{2} \cdot \frac{1}{{x}^{2}}\right)} \]

       7. unpow2 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right) + {x}^{2} \cdot \frac{1}{{x}^{2}}\right) \]

       8. associate-*l* N/A

          \[\leadsto x \cdot \left(\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right)\right)} + {x}^{2} \cdot \frac{1}{{x}^{2}}\right) \]

       9. *-commutative N/A

          \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right) \cdot x\right)} + {x}^{2} \cdot \frac{1}{{x}^{2}}\right) \]

       10. rgt-mult-inverse N/A

           \[\leadsto x \cdot \left(x \cdot \left(\left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right) \cdot x\right) + \color{blue}{1}\right) \]

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

           \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right) \cdot x, 1\right)} \]

   13. Simplified 63.7%

       \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.125, 0.25\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 66.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(x, 0.125, 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.5) 1.0 (* x (* x (fma x 0.125 0.25)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.5) {
		tmp = 1.0;
	} else {
		tmp = x * (x * fma(x, 0.125, 0.25));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.5)
		tmp = 1.0;
	else
		tmp = Float64(x * Float64(x * fma(x, 0.125, 0.25)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.5], 1.0, N[(x * N[(x * N[(x * 0.125 + 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   1. Initial program 35.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 68.6%

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

   if 1.5 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot x + 1} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{2}} + 1 \]

      3. accelerator-lowering-fma.f64 6.0

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

   5. Simplified 6.0%

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

      1. flip-+ N/A

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

      2. metadata-eval N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. flip3-- N/A

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

      6. associate-/r/ N/A

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

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

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

   7. Applied egg-rr 1.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x \cdot \left(x \cdot 0.25\right)}{\mathsf{fma}\left(0.125, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.25, 0.5\right), 1\right), -\frac{1}{\mathsf{fma}\left(x, 0.5, -1\right)}\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \color{blue}{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot x\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot x\right)\right) + 1}\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot x\right), 1\right)}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot x\right) + \frac{1}{2}}, 1\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{4} + \frac{1}{8} \cdot x, \frac{1}{2}\right)}, 1\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{8} \cdot x + \frac{1}{4}}, \frac{1}{2}\right), 1\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{8}} + \frac{1}{4}, \frac{1}{2}\right), 1\right)\right) \]

      7. accelerator-lowering-fma.f64 8.0

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot 0.25\right)}{\mathsf{fma}\left(0.125, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.25, 0.5\right), 1\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.125, 0.25\right)}, 0.5\right), 1\right)\right) \]

   10. Simplified 8.0%

       \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot 0.25\right)}{\mathsf{fma}\left(0.125, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.25, 0.5\right), 1\right), \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.125, 0.25\right), 0.5\right), 1\right)}\right) \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{{x}^{3} \cdot \left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right)} \]
   12. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right) \]

       2. unpow2 N/A

          \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right) \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right)\right)} \]

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

          \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right)\right)\right)} \]

       7. *-commutative N/A

          \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right) \cdot x\right)}\right) \]

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

          \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right) \cdot x\right)\right)} \]

       9. *-commutative N/A

          \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{8} + \frac{1}{4} \cdot \frac{1}{x}\right)\right)}\right) \]

       10. distribute-lft-in N/A

           \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{8} + x \cdot \left(\frac{1}{4} \cdot \frac{1}{x}\right)\right)}\right) \]

       11. *-commutative N/A

           \[\leadsto x \cdot \left(x \cdot \left(x \cdot \frac{1}{8} + x \cdot \color{blue}{\left(\frac{1}{x} \cdot \frac{1}{4}\right)}\right)\right) \]

       12. associate-*r* N/A

           \[\leadsto x \cdot \left(x \cdot \left(x \cdot \frac{1}{8} + \color{blue}{\left(x \cdot \frac{1}{x}\right) \cdot \frac{1}{4}}\right)\right) \]

       13. rgt-mult-inverse N/A

           \[\leadsto x \cdot \left(x \cdot \left(x \cdot \frac{1}{8} + \color{blue}{1} \cdot \frac{1}{4}\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto x \cdot \left(x \cdot \left(x \cdot \frac{1}{8} + \color{blue}{\frac{1}{4}}\right)\right) \]

       15. accelerator-lowering-fma.f64 63.7

           \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, 0.125, 0.25\right)}\right) \]

   13. Simplified 63.7%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x, 0.125, 0.25\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 66.9% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.125\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 2.0) 1.0 (* (* x (* x x)) 0.125)))

C

double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = (x * (x * x)) * 0.125;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = (x * (x * x)) * 0.125d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = (x * (x * x)) * 0.125;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.0:
		tmp = 1.0
	else:
		tmp = (x * (x * x)) * 0.125
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x * Float64(x * x)) * 0.125);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = 1.0;
	else
		tmp = (x * (x * x)) * 0.125;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.0], 1.0, N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.125), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 35.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 68.6%

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

   if 2 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot x + 1} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{2}} + 1 \]

      3. accelerator-lowering-fma.f64 6.0

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

   5. Simplified 6.0%

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

      1. flip-+ N/A

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

      2. metadata-eval N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. flip3-- N/A

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

      6. associate-/r/ N/A

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

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

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

   7. Applied egg-rr 1.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x \cdot \left(x \cdot 0.25\right)}{\mathsf{fma}\left(0.125, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.25, 0.5\right), 1\right), -\frac{1}{\mathsf{fma}\left(x, 0.5, -1\right)}\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \color{blue}{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot x\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot x\right)\right) + 1}\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} + x \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot x\right), 1\right)}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{4} + \frac{1}{8} \cdot x\right) + \frac{1}{2}}, 1\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{4} + \frac{1}{8} \cdot x, \frac{1}{2}\right)}, 1\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{8} \cdot x + \frac{1}{4}}, \frac{1}{2}\right), 1\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot \frac{1}{4}\right)}{\mathsf{fma}\left(\frac{1}{8}, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{4}, \frac{1}{2}\right), 1\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{8}} + \frac{1}{4}, \frac{1}{2}\right), 1\right)\right) \]

      7. accelerator-lowering-fma.f64 8.0

         \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot 0.25\right)}{\mathsf{fma}\left(0.125, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.25, 0.5\right), 1\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.125, 0.25\right)}, 0.5\right), 1\right)\right) \]

   10. Simplified 8.0%

       \[\leadsto \mathsf{fma}\left(\frac{x \cdot \left(x \cdot 0.25\right)}{\mathsf{fma}\left(0.125, x \cdot \left(x \cdot x\right), -1\right)}, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.25, 0.5\right), 1\right), \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.125, 0.25\right), 0.5\right), 1\right)}\right) \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{8} \cdot {x}^{3}} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{{x}^{3} \cdot \frac{1}{8}} \]

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

          \[\leadsto \color{blue}{{x}^{3} \cdot \frac{1}{8}} \]

       3. cube-mult N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \frac{1}{8} \]

       4. unpow2 N/A

          \[\leadsto \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \frac{1}{8} \]

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

          \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right)} \cdot \frac{1}{8} \]

       6. unpow2 N/A

          \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \frac{1}{8} \]

       7. *-lowering-*.f64 63.7

          \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot 0.125 \]

   13. Simplified 63.7%

       \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.125} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 66.9% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.9:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.9) 1.0 (* x (* x (* x 0.041666666666666664)))))

C

double code(double x) {
	double tmp;
	if (x <= 2.9) {
		tmp = 1.0;
	} else {
		tmp = x * (x * (x * 0.041666666666666664));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.9d0) then
        tmp = 1.0d0
    else
        tmp = x * (x * (x * 0.041666666666666664d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.9) {
		tmp = 1.0;
	} else {
		tmp = x * (x * (x * 0.041666666666666664));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.9:
		tmp = 1.0
	else:
		tmp = x * (x * (x * 0.041666666666666664))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.9)
		tmp = 1.0;
	else
		tmp = Float64(x * Float64(x * Float64(x * 0.041666666666666664)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.9)
		tmp = 1.0;
	else
		tmp = x * (x * (x * 0.041666666666666664));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.9], 1.0, N[(x * N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.89999999999999991

   1. Initial program 35.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 68.6%

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

   if 2.89999999999999991 < x

   1. Initial program 100.0%

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

      1. accelerator-lowering-expm1.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}}{x} \]
   6. Step-by-step derivation

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

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

      2. +-commutative N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 74.1

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)}, 0.5\right), 1\right)}{x} \]

   7. Simplified 74.1%

      \[\leadsto \frac{\color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right)}}{x} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{3}} \]
   9. Step-by-step derivation

      1. unpow3 N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 63.7

          \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(x \cdot 0.041666666666666664\right)}\right) \]

   10. Simplified 63.7%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(x \cdot 0.041666666666666664\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 66.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma x (fma x (fma x 0.041666666666666664 0.16666666666666666) 0.5) 1.0))

C

double code(double x) {
	return fma(x, fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5), 1.0);
}

Julia

function code(x)
	return fma(x, fma(x, fma(x, 0.041666666666666664, 0.16666666666666666), 0.5), 1.0)
end

Wolfram

code[x_] := N[(x * N[(x * N[(x * 0.041666666666666664 + 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 53.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

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

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

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

   3. +-commutative N/A

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

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

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. accelerator-lowering-fma.f64 66.9

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)}, 0.5\right), 1\right) \]

5. Simplified 66.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right), 0.5\right), 1\right)} \]
6. Add Preprocessing

Alternative 13: 63.1% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.4) 1.0 (* x (fma x 0.16666666666666666 0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = 1.0;
	} else {
		tmp = x * fma(x, 0.16666666666666666, 0.5);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.4)
		tmp = 1.0;
	else
		tmp = Float64(x * fma(x, 0.16666666666666666, 0.5));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.4], 1.0, N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 35.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 68.6%

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

   if 1.3999999999999999 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 57.8

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right) \]

   5. Simplified 57.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)} \]

   6. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*l* N/A

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

      6. lft-mult-inverse N/A

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

      7. metadata-eval N/A

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

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

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. accelerator-lowering-fma.f64 57.8

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

   8. Simplified 57.8%

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

Alternative 14: 63.1% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.4) 1.0 (* x (* x 0.16666666666666666))))

C

double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = 1.0;
	} else {
		tmp = x * (x * 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.4d0) then
        tmp = 1.0d0
    else
        tmp = x * (x * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = 1.0;
	} else {
		tmp = x * (x * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.4:
		tmp = 1.0
	else:
		tmp = x * (x * 0.16666666666666666)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.4)
		tmp = 1.0;
	else
		tmp = Float64(x * Float64(x * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.4)
		tmp = 1.0;
	else
		tmp = x * (x * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.4], 1.0, N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   1. Initial program 35.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 68.6%

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

   if 2.39999999999999991 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 57.8

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right) \]

   5. Simplified 57.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 57.8

         \[\leadsto x \cdot \color{blue}{\left(x \cdot 0.16666666666666666\right)} \]

   8. Simplified 57.8%

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

Alternative 15: 63.2% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x (fma x 0.16666666666666666 0.5) 1.0))

C

double code(double x) {
	return fma(x, fma(x, 0.16666666666666666, 0.5), 1.0);
}

Julia

function code(x)
	return fma(x, fma(x, 0.16666666666666666, 0.5), 1.0)
end

Wolfram

code[x_] := N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 53.3%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 65.6

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)}, 1\right) \]

5. Simplified 65.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right)} \]
6. Add Preprocessing

Alternative 16: 51.4% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 0.5, 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x 0.5 1.0))

C

double code(double x) {
	return fma(x, 0.5, 1.0);
}

Julia

function code(x)
	return fma(x, 0.5, 1.0)
end

Wolfram

code[x_] := N[(x * 0.5 + 1.0), $MachinePrecision]

Derivation

1. Initial program 53.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot x} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + 1} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{x \cdot \frac{1}{2}} + 1 \]

   3. accelerator-lowering-fma.f64 50.9

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

5. Simplified 50.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)} \]
6. Add Preprocessing

Alternative 17: 51.3% accurate, 115.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 53.3%

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

3. Taylor expanded in x around 0

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

5. Simplified 50.7%

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

Alternative 18: 3.3% accurate, 115.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 53.3%

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

3. Taylor expanded in x around 0

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

5. Simplified 3.3%

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

   1. metadata-eval N/A

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

   2. div0 3.3

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

7. Applied egg-rr 3.3%

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

Developer Target 1: 52.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x} - 1\\ \mathbf{if}\;x < 1 \land x > -1:\\ \;\;\;\;\frac{t\_0}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (exp x) 1.0)))
   (if (and (< x 1.0) (> x -1.0)) (/ t_0 (log (exp x))) (/ t_0 x))))

C

double code(double x) {
	double t_0 = exp(x) - 1.0;
	double tmp;
	if ((x < 1.0) && (x > -1.0)) {
		tmp = t_0 / log(exp(x));
	} else {
		tmp = t_0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(x) - 1.0d0
    if ((x < 1.0d0) .and. (x > (-1.0d0))) then
        tmp = t_0 / log(exp(x))
    else
        tmp = t_0 / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.exp(x) - 1.0;
	double tmp;
	if ((x < 1.0) && (x > -1.0)) {
		tmp = t_0 / Math.log(Math.exp(x));
	} else {
		tmp = t_0 / x;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.exp(x) - 1.0
	tmp = 0
	if (x < 1.0) and (x > -1.0):
		tmp = t_0 / math.log(math.exp(x))
	else:
		tmp = t_0 / x
	return tmp

Julia

function code(x)
	t_0 = Float64(exp(x) - 1.0)
	tmp = 0.0
	if ((x < 1.0) && (x > -1.0))
		tmp = Float64(t_0 / log(exp(x)));
	else
		tmp = Float64(t_0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = exp(x) - 1.0;
	tmp = 0.0;
	if ((x < 1.0) && (x > -1.0))
		tmp = t_0 / log(exp(x));
	else
		tmp = t_0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision]}, If[And[Less[x, 1.0], Greater[x, -1.0]], N[(t$95$0 / N[Log[N[Exp[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 / x), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024197 
(FPCore (x)
  :name "Kahan's exp quotient"
  :precision binary64

  :alt
  (! :herbie-platform default (if (and (< x 1) (> x -1)) (/ (- (exp x) 1) (log (exp x))) (/ (- (exp x) 1) x)))

  (/ (- (exp x) 1.0) x))