expm1 (example 3.7)

Percentage Accurate: 8.2% → 100.0%

Time: 6.1s

Alternatives: 8

Speedup: 104.0×

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: 8.2% 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 7.8%

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

   1. accelerator-lowering-expm1.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 99.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{1}{x \cdot \left(\frac{2}{x} + \mathsf{fma}\left(x, 0.05555555555555555, -0.6666666666666666\right)\right)}, x \cdot x, x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (/ 1.0 (* x (+ (/ 2.0 x) (fma x 0.05555555555555555 -0.6666666666666666))))
  (* x x)
  x))

C

double code(double x) {
	return fma((1.0 / (x * ((2.0 / x) + fma(x, 0.05555555555555555, -0.6666666666666666)))), (x * x), x);
}

Julia

function code(x)
	return fma(Float64(1.0 / Float64(x * Float64(Float64(2.0 / x) + fma(x, 0.05555555555555555, -0.6666666666666666)))), Float64(x * x), x)
end

Wolfram

code[x_] := N[(N[(1.0 / N[(x * N[(N[(2.0 / x), $MachinePrecision] + N[(x * 0.05555555555555555 + -0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 7.8%

   \[e^{x} - 1 \]
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

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

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

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

   7. +-commutative N/A

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

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

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

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

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

   12. *-lowering-*.f64 99.4

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

5. Simplified 99.4%

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

   1. flip3-+ N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{\left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right)}^{3} + {\frac{1}{2}}^{3}}{\left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right) \cdot \frac{1}{2}\right)}}, x \cdot x, x\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{\left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right) \cdot \frac{1}{2}\right)}{{\left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right)}^{3} + {\frac{1}{2}}^{3}}}}, x \cdot x, x\right) \]

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

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{\left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right) \cdot \frac{1}{2}\right)}{{\left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right)}^{3} + {\frac{1}{2}}^{3}}}}, x \cdot x, x\right) \]

   4. clear-num N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{\frac{1}{\frac{{\left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right)}^{3} + {\frac{1}{2}}^{3}}{\left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right) \cdot \frac{1}{2}\right)}}}}, x \cdot x, x\right) \]

   5. flip3-+ N/A

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

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

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

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

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

   8. accelerator-lowering-fma.f64 99.4

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

7. Applied egg-rr 99.4%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{2 + x \cdot \left(\frac{1}{18} \cdot x - \frac{2}{3}\right)}}, x \cdot x, x\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{x \cdot \left(\frac{1}{18} \cdot x - \frac{2}{3}\right) + 2}}, x \cdot x, x\right) \]

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

      \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{\mathsf{fma}\left(x, \frac{1}{18} \cdot x - \frac{2}{3}, 2\right)}}, x \cdot x, x\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(x, \color{blue}{\frac{1}{18} \cdot x + \left(\mathsf{neg}\left(\frac{2}{3}\right)\right)}, 2\right)}, x \cdot x, x\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{18}} + \left(\mathsf{neg}\left(\frac{2}{3}\right)\right), 2\right)}, x \cdot x, x\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(x, x \cdot \frac{1}{18} + \color{blue}{\frac{-2}{3}}, 2\right)}, x \cdot x, x\right) \]

   6. accelerator-lowering-fma.f64 99.5

      \[\leadsto \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.05555555555555555, -0.6666666666666666\right)}, 2\right)}, x \cdot x, x\right) \]

10. Simplified 99.5%

    \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.05555555555555555, -0.6666666666666666\right), 2\right)}}, x \cdot x, x\right) \]

11. Taylor expanded in x around inf

    \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{{x}^{2} \cdot \left(\left(\frac{1}{18} + \frac{2}{{x}^{2}}\right) - \frac{2}{3} \cdot \frac{1}{x}\right)}}, x \cdot x, x\right) \]

13. Simplified 99.5%

    \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{x \cdot \left(\frac{2}{x} + \mathsf{fma}\left(x, 0.05555555555555555, -0.6666666666666666\right)\right)}}, x \cdot x, x\right) \]
14. Add Preprocessing

Alternative 3: 99.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.05555555555555555, -0.6666666666666666\right), 2\right)}, x, x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (/ x (fma x (fma x 0.05555555555555555 -0.6666666666666666) 2.0)) x x))

C

double code(double x) {
	return fma((x / fma(x, fma(x, 0.05555555555555555, -0.6666666666666666), 2.0)), x, x);
}

Julia

function code(x)
	return fma(Float64(x / fma(x, fma(x, 0.05555555555555555, -0.6666666666666666), 2.0)), x, x)
end

Wolfram

code[x_] := N[(N[(x / N[(x * N[(x * 0.05555555555555555 + -0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]

Derivation

1. Initial program 7.8%

   \[e^{x} - 1 \]
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

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

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

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

   7. +-commutative N/A

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

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

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

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

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

   12. *-lowering-*.f64 99.4

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

5. Simplified 99.4%

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

   1. flip3-+ N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{{\left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right)}^{3} + {\frac{1}{2}}^{3}}{\left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right) \cdot \frac{1}{2}\right)}}, x \cdot x, x\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{\left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right) \cdot \frac{1}{2}\right)}{{\left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right)}^{3} + {\frac{1}{2}}^{3}}}}, x \cdot x, x\right) \]

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

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{\left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right) \cdot \frac{1}{2}\right)}{{\left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right)}^{3} + {\frac{1}{2}}^{3}}}}, x \cdot x, x\right) \]

   4. clear-num N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{\frac{1}{\frac{{\left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right)}^{3} + {\frac{1}{2}}^{3}}{\left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{1}{6}\right)\right) \cdot \frac{1}{2}\right)}}}}, x \cdot x, x\right) \]

   5. flip3-+ N/A

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

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

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

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

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

   8. accelerator-lowering-fma.f64 99.4

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

7. Applied egg-rr 99.4%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{2 + x \cdot \left(\frac{1}{18} \cdot x - \frac{2}{3}\right)}}, x \cdot x, x\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{x \cdot \left(\frac{1}{18} \cdot x - \frac{2}{3}\right) + 2}}, x \cdot x, x\right) \]

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

      \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{\mathsf{fma}\left(x, \frac{1}{18} \cdot x - \frac{2}{3}, 2\right)}}, x \cdot x, x\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(x, \color{blue}{\frac{1}{18} \cdot x + \left(\mathsf{neg}\left(\frac{2}{3}\right)\right)}, 2\right)}, x \cdot x, x\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{18}} + \left(\mathsf{neg}\left(\frac{2}{3}\right)\right), 2\right)}, x \cdot x, x\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(x, x \cdot \frac{1}{18} + \color{blue}{\frac{-2}{3}}, 2\right)}, x \cdot x, x\right) \]

   6. accelerator-lowering-fma.f64 99.5

      \[\leadsto \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.05555555555555555, -0.6666666666666666\right)}, 2\right)}, x \cdot x, x\right) \]

10. Simplified 99.5%

    \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.05555555555555555, -0.6666666666666666\right), 2\right)}}, x \cdot x, x\right) \]
11. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\frac{1}{x \cdot \left(x \cdot \frac{1}{18} + \frac{-2}{3}\right) + 2} \cdot x\right) \cdot x} + x \]

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

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{x \cdot \left(x \cdot \frac{1}{18} + \frac{-2}{3}\right) + 2} \cdot x, x, x\right)} \]

    3. associate-*l/ N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1 \cdot x}{x \cdot \left(x \cdot \frac{1}{18} + \frac{-2}{3}\right) + 2}}, x, x\right) \]

    4. *-lft-identity N/A

       \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x}}{x \cdot \left(x \cdot \frac{1}{18} + \frac{-2}{3}\right) + 2}, x, x\right) \]

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

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{x \cdot \left(x \cdot \frac{1}{18} + \frac{-2}{3}\right) + 2}}, x, x\right) \]

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

       \[\leadsto \mathsf{fma}\left(\frac{x}{\color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{18} + \frac{-2}{3}, 2\right)}}, x, x\right) \]

    7. accelerator-lowering-fma.f64 99.5

       \[\leadsto \mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.05555555555555555, -0.6666666666666666\right)}, 2\right)}, x, x\right) \]

12. Applied egg-rr 99.5%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.05555555555555555, -0.6666666666666666\right), 2\right)}, x, x\right)} \]
13. Add Preprocessing

Alternative 4: 99.6% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.8%

   \[e^{x} - 1 \]
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

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

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

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

   7. +-commutative N/A

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

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

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

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

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

   12. *-lowering-*.f64 99.4

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

5. Simplified 99.4%

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

Alternative 5: 99.5% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.8%

   \[e^{x} - 1 \]
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

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

   2. distribute-rgt-in N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. *-lft-identity N/A

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

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

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

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

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

   10. *-lowering-*.f64 99.2

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

5. Simplified 99.2%

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

Alternative 6: 99.1% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.8%

   \[e^{x} - 1 \]
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

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

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

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

   5. *-commutative N/A

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

   6. *-lowering-*.f64 99.0

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

5. Simplified 99.0%

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

Alternative 7: 99.1% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.8%

   \[e^{x} - 1 \]
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

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

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

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

   5. *-commutative N/A

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

   6. *-lowering-*.f64 99.0

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

5. Simplified 99.0%

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

   1. *-commutative N/A

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

   2. distribute-lft1-in N/A

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

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

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

   4. accelerator-lowering-fma.f64 99.0

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

7. Applied egg-rr 99.0%

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

8. Final simplification 99.0%

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

Alternative 8: 98.2% accurate, 104.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 7.8%

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

3. Taylor expanded in x around 0

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

5. Simplified 98.3%

   \[\leadsto \color{blue}{x} \]
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 2024205 
(FPCore (x)
  :name "expm1 (example 3.7)"
  :precision binary64
  :pre (<= (fabs x) 1.0)

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

  (- (exp x) 1.0))