expm1 (example 3.7)

Percentage Accurate: 8.4% → 100.0%

Time: 4.4s

Alternatives: 9

Speedup: 17.3×

Specification

\[\left|x\right| \leq 1\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 8.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(x)
	return expm1(x)
end

Wolfram

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

Derivation

1. Initial program 9.2%

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

   1. lift--.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lower-expm1.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 99.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.08333333333333333, x, -0.5\right) \cdot x\\ \frac{x}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x, -0.5\right), x, -1\right)} \cdot \mathsf{fma}\left(t\_0, t\_0, -1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fma 0.08333333333333333 x -0.5) x)))
   (/
    x
    (*
     (/ 1.0 (fma (fma 0.08333333333333333 x -0.5) x -1.0))
     (fma t_0 t_0 -1.0)))))

C

double code(double x) {
	double t_0 = fma(0.08333333333333333, x, -0.5) * x;
	return x / ((1.0 / fma(fma(0.08333333333333333, x, -0.5), x, -1.0)) * fma(t_0, t_0, -1.0));
}

Julia

function code(x)
	t_0 = Float64(fma(0.08333333333333333, x, -0.5) * x)
	return Float64(x / Float64(Float64(1.0 / fma(fma(0.08333333333333333, x, -0.5), x, -1.0)) * fma(t_0, t_0, -1.0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(0.08333333333333333 * x + -0.5), $MachinePrecision] * x), $MachinePrecision]}, N[(x / N[(N[(1.0 / N[(N[(0.08333333333333333 * x + -0.5), $MachinePrecision] * x + -1.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 9.2%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 99.4

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

5. Applied rewrites 99.4%

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.5%

    \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x, -0.5\right), \color{blue}{x}, 1\right)} \]
11. Step-by-step derivation

12. Applied rewrites 99.5%

    \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x, -0.5\right) \cdot x, \mathsf{fma}\left(0.08333333333333333, x, -0.5\right) \cdot x, -1\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x, -0.5\right), x, -1\right)}}} \]

13. Final simplification 99.5%

    \[\leadsto \frac{x}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x, -0.5\right), x, -1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x, -0.5\right) \cdot x, \mathsf{fma}\left(0.08333333333333333, x, -0.5\right) \cdot x, -1\right)} \]
14. Add Preprocessing

Alternative 3: 99.6% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x, -0.5\right), x, 1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ x (fma (fma 0.08333333333333333 x -0.5) x 1.0)))

C

double code(double x) {
	return x / fma(fma(0.08333333333333333, x, -0.5), x, 1.0);
}

Julia

function code(x)
	return Float64(x / fma(fma(0.08333333333333333, x, -0.5), x, 1.0))
end

Wolfram

code[x_] := N[(x / N[(N[(0.08333333333333333 * x + -0.5), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 9.2%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 99.4

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

5. Applied rewrites 99.4%

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.5%

    \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x, -0.5\right), \color{blue}{x}, 1\right)} \]
11. Add Preprocessing

Alternative 4: 99.6% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.2%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 99.4

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

5. Applied rewrites 99.4%

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

Alternative 5: 99.4% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.2%

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

   1. lift--.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lower-expm1.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 99.2

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

7. Applied rewrites 99.2%

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

Alternative 6: 99.4% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.2%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 99.2

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

5. Applied rewrites 99.2%

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

Alternative 7: 99.1% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.2%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 98.6

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

5. Applied rewrites 98.6%

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

Alternative 8: 98.0% accurate, 17.3× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* 1.0 x))

C

double code(double x) {
	return 1.0 * x;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0 * x

Julia

function code(x)
	return Float64(1.0 * x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.2%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

   \[\leadsto 1 \cdot x \]
7. Step-by-step derivation

8. Applied rewrites 97.3%

   \[\leadsto 1 \cdot x \]
9. Add Preprocessing

Alternative 9: 5.3% accurate, 26.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 - 1.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 9.2%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 5.3%

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

Developer Target 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(x)
	return expm1(x)
end

Wolfram

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

Reproduce

herbie shell --seed 2024240 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :precision binary64
  :pre (<= (fabs x) 1.0)

  :alt
  (! :herbie-platform default (expm1 x))

  (- (exp x) 1.0))