Kahan's exp quotient

Percentage Accurate: 53.2% → 100.0%

Time: 3.8s

Alternatives: 18

Speedup: 9.6×

• 25.0% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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: 53.2% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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 54.1%

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

   1. lift--.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lower-expm1.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 69.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{x} - 1}{x} \leq 1.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.027777777777777776}{\mathsf{fma}\left(0.041666666666666664, x, -0.16666666666666666\right)}, x, 0.5\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, x, 1\right) \cdot x}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (- (exp x) 1.0) x) 1.2)
   (fma
    (fma
     (/
      -0.027777777777777776
      (fma 0.041666666666666664 x -0.16666666666666666))
     x
     0.5)
    x
    1.0)
   (/ (* (fma (* (* x x) 0.041666666666666664) x 1.0) x) x)))

C

double code(double x) {
	double tmp;
	if (((exp(x) - 1.0) / x) <= 1.2) {
		tmp = fma(fma((-0.027777777777777776 / fma(0.041666666666666664, x, -0.16666666666666666)), x, 0.5), x, 1.0);
	} else {
		tmp = (fma(((x * x) * 0.041666666666666664), x, 1.0) * x) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(exp(x) - 1.0) / x) <= 1.2)
		tmp = fma(fma(Float64(-0.027777777777777776 / fma(0.041666666666666664, x, -0.16666666666666666)), x, 0.5), x, 1.0);
	else
		tmp = Float64(Float64(fma(Float64(Float64(x * x) * 0.041666666666666664), x, 1.0) * x) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision] / x), $MachinePrecision], 1.2], N[(N[(N[(-0.027777777777777776 / N[(0.041666666666666664 * x + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (exp.f64 x) #s(literal 1 binary64)) x) < 1.19999999999999996

   1. Initial program 34.4%

      \[\frac{e^{x} - 1}{x} \]
   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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 71.7

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

   5. Applied rewrites 71.7%

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

      1. lift-fma.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. swap-sqr N/A

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

      6. pow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. fp-cancel-sub-sign-inv N/A

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

      15. distribute-lft-neg-in N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. metadata-eval 71.7

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

   7. Applied rewrites 71.7%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{-1}{36}}{\mathsf{fma}\left(\frac{1}{24}, x, \frac{-1}{6}\right)}, x, \frac{1}{2}\right), x, 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 72.3%

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

   if 1.19999999999999996 < (/.f64 (-.f64 (exp.f64 x) #s(literal 1 binary64)) x)

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 75.5

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2}, x, 1\right) \cdot x}{x} \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 75.5

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

   8. Applied rewrites 75.5%

      \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, x, 1\right) \cdot x}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 67.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{x} - 1}{x} \leq 1.2:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(x \cdot x, 0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (- (exp x) 1.0) x) 1.2)
   (fma x 0.5 (fma (* x x) 0.16666666666666666 1.0))
   (fma (* (* x x) 0.041666666666666664) x 1.0)))

C

double code(double x) {
	double tmp;
	if (((exp(x) - 1.0) / x) <= 1.2) {
		tmp = fma(x, 0.5, fma((x * x), 0.16666666666666666, 1.0));
	} else {
		tmp = fma(((x * x) * 0.041666666666666664), x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(exp(x) - 1.0) / x) <= 1.2)
		tmp = fma(x, 0.5, fma(Float64(x * x), 0.16666666666666666, 1.0));
	else
		tmp = fma(Float64(Float64(x * x) * 0.041666666666666664), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision] / x), $MachinePrecision], 1.2], N[(x * 0.5 + N[(N[(x * x), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * x + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (exp.f64 x) #s(literal 1 binary64)) x) < 1.19999999999999996

   1. Initial program 34.4%

      \[\frac{e^{x} - 1}{x} \]
   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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 71.8

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

   5. Applied rewrites 71.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. pow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 71.8

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

   7. Applied rewrites 71.8%

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

   if 1.19999999999999996 < (/.f64 (-.f64 (exp.f64 x) #s(literal 1 binary64)) x)

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 67.3

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2}, x, 1\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 67.3

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

   8. Applied rewrites 67.3%

      \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, x, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 67.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{x} - 1}{x} \leq 1.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (- (exp x) 1.0) x) 1.2)
   (fma (fma 0.16666666666666666 x 0.5) x 1.0)
   (fma (* (* x x) 0.041666666666666664) x 1.0)))

C

double code(double x) {
	double tmp;
	if (((exp(x) - 1.0) / x) <= 1.2) {
		tmp = fma(fma(0.16666666666666666, x, 0.5), x, 1.0);
	} else {
		tmp = fma(((x * x) * 0.041666666666666664), x, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(exp(x) - 1.0) / x) <= 1.2)
		tmp = fma(fma(0.16666666666666666, x, 0.5), x, 1.0);
	else
		tmp = fma(Float64(Float64(x * x) * 0.041666666666666664), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision] / x), $MachinePrecision], 1.2], N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * x + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (exp.f64 x) #s(literal 1 binary64)) x) < 1.19999999999999996

   1. Initial program 34.4%

      \[\frac{e^{x} - 1}{x} \]
   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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 71.8

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

   5. Applied rewrites 71.8%

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

   if 1.19999999999999996 < (/.f64 (-.f64 (exp.f64 x) #s(literal 1 binary64)) x)

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 67.3

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2}, x, 1\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 67.3

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

   8. Applied rewrites 67.3%

      \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, x, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 67.9% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (- (exp x) 1.0) x) 2.0)
   (fma (fma 0.16666666666666666 x 0.5) x 1.0)
   (* (* (fma 0.041666666666666664 x 0.16666666666666666) x) x)))

C

double code(double x) {
	double tmp;
	if (((exp(x) - 1.0) / x) <= 2.0) {
		tmp = fma(fma(0.16666666666666666, x, 0.5), x, 1.0);
	} else {
		tmp = (fma(0.041666666666666664, x, 0.16666666666666666) * x) * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(exp(x) - 1.0) / x) <= 2.0)
		tmp = fma(fma(0.16666666666666666, x, 0.5), x, 1.0);
	else
		tmp = Float64(Float64(fma(0.041666666666666664, x, 0.16666666666666666) * x) * x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision] / x), $MachinePrecision], 2.0], N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision], N[(N[(N[(0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (exp.f64 x) #s(literal 1 binary64)) x) < 2

   1. Initial program 34.4%

      \[\frac{e^{x} - 1}{x} \]
   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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 71.9

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

   5. Applied rewrites 71.9%

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

   if 2 < (/.f64 (-.f64 (exp.f64 x) #s(literal 1 binary64)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 66.9

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

   5. Applied rewrites 66.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. associate-*r* N/A

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

      17. pow2 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-fma.f64 66.9

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

   7. Applied rewrites 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. flip-+ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. fp-cancel-sub-sign-inv N/A

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

      11. swap-sqr N/A

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

      12. pow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. metadata-eval N/A

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

      22. fp-cancel-sub-sign-inv N/A

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

      23. distribute-lft-neg-in N/A

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

   9. Applied rewrites 75.2%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 66.9%

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

Alternative 6: 63.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{x} - 1}{x} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (- (exp x) 1.0) x) 2.0)
   (fma 0.5 x 1.0)
   (* (* x x) 0.16666666666666666)))

C

double code(double x) {
	double tmp;
	if (((exp(x) - 1.0) / x) <= 2.0) {
		tmp = fma(0.5, x, 1.0);
	} else {
		tmp = (x * x) * 0.16666666666666666;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(exp(x) - 1.0) / x) <= 2.0)
		tmp = fma(0.5, x, 1.0);
	else
		tmp = Float64(Float64(x * x) * 0.16666666666666666);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision] / x), $MachinePrecision], 2.0], N[(0.5 * x + 1.0), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (exp.f64 x) #s(literal 1 binary64)) x) < 2

   1. Initial program 34.4%

      \[\frac{e^{x} - 1}{x} \]
   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

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

      2. lower-fma.f64 71.0

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

   5. Applied rewrites 71.0%

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

   if 2 < (/.f64 (-.f64 (exp.f64 x) #s(literal 1 binary64)) 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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 53.2

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

   5. Applied rewrites 53.2%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto {x}^{2} \cdot \frac{1}{6} \]

      2. lower-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \frac{1}{6} \]

      3. pow2 N/A

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

      4. lift-*.f64 53.2

         \[\leadsto \left(x \cdot x\right) \cdot 0.16666666666666666 \]

   8. Applied rewrites 53.2%

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

Alternative 7: 72.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right)\\ \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.027777777777777776}{\mathsf{fma}\left(0.041666666666666664, x, -0.16666666666666666\right)}, x, 0.5\right), x, 1\right)\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{x \cdot x - t\_0 \cdot t\_0}{x - t\_0}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, x, 1\right) \cdot x}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (*
          (* x x)
          (fma (fma 0.041666666666666664 x 0.16666666666666666) x 0.5))))
   (if (<= x 2.5)
     (fma
      (fma
       (/
        -0.027777777777777776
        (fma 0.041666666666666664 x -0.16666666666666666))
       x
       0.5)
      x
      1.0)
     (if (<= x 2.6e+77)
       (/ (/ (- (* x x) (* t_0 t_0)) (- x t_0)) x)
       (/ (* (fma (* (* x x) 0.041666666666666664) x 1.0) x) x)))))

C

double code(double x) {
	double t_0 = (x * x) * fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5);
	double tmp;
	if (x <= 2.5) {
		tmp = fma(fma((-0.027777777777777776 / fma(0.041666666666666664, x, -0.16666666666666666)), x, 0.5), x, 1.0);
	} else if (x <= 2.6e+77) {
		tmp = (((x * x) - (t_0 * t_0)) / (x - t_0)) / x;
	} else {
		tmp = (fma(((x * x) * 0.041666666666666664), x, 1.0) * x) / x;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5))
	tmp = 0.0
	if (x <= 2.5)
		tmp = fma(fma(Float64(-0.027777777777777776 / fma(0.041666666666666664, x, -0.16666666666666666)), x, 0.5), x, 1.0);
	elseif (x <= 2.6e+77)
		tmp = Float64(Float64(Float64(Float64(x * x) - Float64(t_0 * t_0)) / Float64(x - t_0)) / x);
	else
		tmp = Float64(Float64(fma(Float64(Float64(x * x) * 0.041666666666666664), x, 1.0) * x) / x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(N[(0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * x + 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.5], N[(N[(N[(-0.027777777777777776 / N[(0.041666666666666664 * x + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision], If[LessEqual[x, 2.6e+77], N[(N[(N[(N[(x * x), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x - t$95$0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] / x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.5

   1. Initial program 34.4%

      \[\frac{e^{x} - 1}{x} \]
   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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 71.8

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

   5. Applied rewrites 71.8%

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

      1. lift-fma.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. swap-sqr N/A

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

      6. pow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. fp-cancel-sub-sign-inv N/A

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

      15. distribute-lft-neg-in N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. metadata-eval 71.8

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

   7. Applied rewrites 71.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{-1}{36}}{\mathsf{fma}\left(\frac{1}{24}, x, \frac{-1}{6}\right)}, x, \frac{1}{2}\right), x, 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 72.5%

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

   if 2.5 < x < 2.6000000000000002e77

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 4.5

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

   5. Applied rewrites 4.5%

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

   7. Applied rewrites 56.7%

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

   if 2.6000000000000002e77 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2}, x, 1\right) \cdot x}{x} \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 100.0

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

   8. Applied rewrites 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.041666666666666664, x, 1\right) \cdot x}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 70.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), 1\right)\\ \mathbf{if}\;x \leq 1.65 \cdot 10^{+103}:\\ \;\;\;\;\frac{0.25 \cdot \left(x \cdot x\right) - t\_0 \cdot t\_0}{0.5 \cdot x - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right) \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (fma (* x x) (fma 0.041666666666666664 x 0.16666666666666666) 1.0)))
   (if (<= x 1.65e+103)
     (/ (- (* 0.25 (* x x)) (* t_0 t_0)) (- (* 0.5 x) t_0))
     (* (* (fma 0.041666666666666664 x 0.16666666666666666) x) x))))

C

double code(double x) {
	double t_0 = fma((x * x), fma(0.041666666666666664, x, 0.16666666666666666), 1.0);
	double tmp;
	if (x <= 1.65e+103) {
		tmp = ((0.25 * (x * x)) - (t_0 * t_0)) / ((0.5 * x) - t_0);
	} else {
		tmp = (fma(0.041666666666666664, x, 0.16666666666666666) * x) * x;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(Float64(x * x), fma(0.041666666666666664, x, 0.16666666666666666), 1.0)
	tmp = 0.0
	if (x <= 1.65e+103)
		tmp = Float64(Float64(Float64(0.25 * Float64(x * x)) - Float64(t_0 * t_0)) / Float64(Float64(0.5 * x) - t_0));
	else
		tmp = Float64(Float64(fma(0.041666666666666664, x, 0.16666666666666666) * x) * x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, 1.65e+103], N[(N[(N[(0.25 * N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * x), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.65000000000000004e103

   1. Initial program 43.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 63.1

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

   5. Applied rewrites 63.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. associate-*r* N/A

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

      17. pow2 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-fma.f64 63.1

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

   7. Applied rewrites 63.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. flip-+ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. fp-cancel-sub-sign-inv N/A

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

      11. swap-sqr N/A

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

      12. pow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. metadata-eval N/A

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

      22. fp-cancel-sub-sign-inv N/A

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

      23. distribute-lft-neg-in N/A

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

   9. Applied rewrites 66.2%

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

   11. Applied rewrites 69.6%

       \[\leadsto \frac{0.25 \cdot \left(x \cdot x\right) - \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), 1\right)}{\color{blue}{0.5 \cdot x - \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), 1\right)}} \]

   if 1.65000000000000004e103 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. associate-*r* N/A

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

      17. pow2 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. flip-+ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. fp-cancel-sub-sign-inv N/A

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

      11. swap-sqr N/A

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

      12. pow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. metadata-eval N/A

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

      22. fp-cancel-sub-sign-inv N/A

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

      23. distribute-lft-neg-in N/A

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 100.0%

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

Alternative 9: 70.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.25, x \cdot x, -1\right), \mathsf{fma}\left(0.041666666666666664, x, -0.16666666666666666\right), \mathsf{fma}\left(0.5, x, -1\right) \cdot \left(\left(\mathsf{fma}\left(x \cdot x, 0.001736111111111111, -0.027777777777777776\right) \cdot x\right) \cdot x\right)\right)}{\mathsf{fma}\left(0.5, x, -1\right) \cdot \mathsf{fma}\left(0.041666666666666664, x, -0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e+153)
   (/
    (fma
     (fma 0.25 (* x x) -1.0)
     (fma 0.041666666666666664 x -0.16666666666666666)
     (*
      (fma 0.5 x -1.0)
      (* (* (fma (* x x) 0.001736111111111111 -0.027777777777777776) x) x)))
    (* (fma 0.5 x -1.0) (fma 0.041666666666666664 x -0.16666666666666666)))
   (* (* x x) 0.16666666666666666)))

C

double code(double x) {
	double tmp;
	if (x <= 5e+153) {
		tmp = fma(fma(0.25, (x * x), -1.0), fma(0.041666666666666664, x, -0.16666666666666666), (fma(0.5, x, -1.0) * ((fma((x * x), 0.001736111111111111, -0.027777777777777776) * x) * x))) / (fma(0.5, x, -1.0) * fma(0.041666666666666664, x, -0.16666666666666666));
	} else {
		tmp = (x * x) * 0.16666666666666666;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e+153)
		tmp = Float64(fma(fma(0.25, Float64(x * x), -1.0), fma(0.041666666666666664, x, -0.16666666666666666), Float64(fma(0.5, x, -1.0) * Float64(Float64(fma(Float64(x * x), 0.001736111111111111, -0.027777777777777776) * x) * x))) / Float64(fma(0.5, x, -1.0) * fma(0.041666666666666664, x, -0.16666666666666666)));
	else
		tmp = Float64(Float64(x * x) * 0.16666666666666666);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5e+153], N[(N[(N[(0.25 * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision] * N[(0.041666666666666664 * x + -0.16666666666666666), $MachinePrecision] + N[(N[(0.5 * x + -1.0), $MachinePrecision] * N[(N[(N[(N[(x * x), $MachinePrecision] * 0.001736111111111111 + -0.027777777777777776), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * x + -1.0), $MachinePrecision] * N[(0.041666666666666664 * x + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000018e153

   1. Initial program 45.8%

      \[\frac{e^{x} - 1}{x} \]
   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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 65.0

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

   5. Applied rewrites 65.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. associate-+r+ N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. +-commutative N/A

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

      16. associate-*r* N/A

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

      17. pow2 N/A

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

      18. lower-*.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-fma.f64 65.0

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

   7. Applied rewrites 65.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. flip-+ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. fp-cancel-sub-sign-inv N/A

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

      11. swap-sqr N/A

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

      12. pow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. metadata-eval N/A

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

      22. fp-cancel-sub-sign-inv N/A

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

      23. distribute-lft-neg-in N/A

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

   9. Applied rewrites 67.9%

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

   11. Applied rewrites 70.3%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.25, x \cdot x, -1\right), \mathsf{fma}\left(0.041666666666666664, x, -0.16666666666666666\right), \mathsf{fma}\left(0.5, x, -1\right) \cdot \left(\left(\mathsf{fma}\left(x \cdot x, 0.001736111111111111, -0.027777777777777776\right) \cdot x\right) \cdot x\right)\right)}{\color{blue}{\mathsf{fma}\left(0.5, x, -1\right) \cdot \mathsf{fma}\left(0.041666666666666664, x, -0.16666666666666666\right)}} \]

   if 5.00000000000000018e153 < 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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto {x}^{2} \cdot \frac{1}{6} \]

      2. lower-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \frac{1}{6} \]

      3. pow2 N/A

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

      4. lift-*.f64 100.0

         \[\leadsto \left(x \cdot x\right) \cdot 0.16666666666666666 \]

   8. Applied rewrites 100.0%

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

Alternative 10: 69.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.5, x, 1\right) + \frac{\mathsf{fma}\left(0.001736111111111111, x \cdot x, -0.027777777777777776\right) \cdot \left(x \cdot x\right)}{\mathsf{fma}\left(0.041666666666666664, x, -0.16666666666666666\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+
  (fma 0.5 x 1.0)
  (/
   (* (fma 0.001736111111111111 (* x x) -0.027777777777777776) (* x x))
   (fma 0.041666666666666664 x -0.16666666666666666))))

C

double code(double x) {
	return fma(0.5, x, 1.0) + ((fma(0.001736111111111111, (x * x), -0.027777777777777776) * (x * x)) / fma(0.041666666666666664, x, -0.16666666666666666));
}

Julia

function code(x)
	return Float64(fma(0.5, x, 1.0) + Float64(Float64(fma(0.001736111111111111, Float64(x * x), -0.027777777777777776) * Float64(x * x)) / fma(0.041666666666666664, x, -0.16666666666666666)))
end

Wolfram

code[x_] := N[(N[(0.5 * x + 1.0), $MachinePrecision] + N[(N[(N[(0.001736111111111111 * N[(x * x), $MachinePrecision] + -0.027777777777777776), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(0.041666666666666664 * x + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

   \[\frac{e^{x} - 1}{x} \]
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

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 70.3

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

5. Applied rewrites 70.3%

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

   1. lift-fma.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. +-commutative N/A

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

   9. distribute-rgt-in N/A

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

   10. associate-+r+ N/A

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

   11. lower-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. +-commutative N/A

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

   16. associate-*r* N/A

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

   17. pow2 N/A

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

   18. lower-*.f64 N/A

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

   19. pow2 N/A

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

   20. lift-*.f64 N/A

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

   21. lift-fma.f64 70.3

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

7. Applied rewrites 70.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. flip-+ N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. fp-cancel-sub-sign-inv N/A

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

   11. swap-sqr N/A

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

   12. pow2 N/A

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

   13. lower-fma.f64 N/A

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

   14. metadata-eval N/A

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

   15. pow2 N/A

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

   16. lift-*.f64 N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. pow2 N/A

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

   20. lift-*.f64 N/A

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

   21. metadata-eval N/A

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

   22. fp-cancel-sub-sign-inv N/A

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

   23. distribute-lft-neg-in N/A

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

9. Applied rewrites 72.8%

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

Alternative 11: 69.3% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.1%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 72.8

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

5. Applied rewrites 72.8%

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

Alternative 12: 67.6% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.1%

   \[\frac{e^{x} - 1}{x} \]
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

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 70.3

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

5. Applied rewrites 70.3%

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

   1. lift-fma.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. +-commutative N/A

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

   9. distribute-rgt-in N/A

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

   10. associate-+r+ N/A

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

   11. lower-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. +-commutative N/A

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

   16. associate-*r* N/A

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

   17. pow2 N/A

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

   18. lower-*.f64 N/A

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

   19. pow2 N/A

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

   20. lift-*.f64 N/A

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

   21. lift-fma.f64 70.3

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

7. Applied rewrites 70.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. flip-+ N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. fp-cancel-sub-sign-inv N/A

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

   11. swap-sqr N/A

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

   12. pow2 N/A

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

   13. lower-fma.f64 N/A

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

   14. metadata-eval N/A

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

   15. pow2 N/A

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

   16. lift-*.f64 N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. pow2 N/A

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

   20. lift-*.f64 N/A

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

   21. metadata-eval N/A

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

   22. fp-cancel-sub-sign-inv N/A

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

   23. distribute-lft-neg-in N/A

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

9. Applied rewrites 72.8%

   \[\leadsto \mathsf{fma}\left(0.5, x, 1\right) + \frac{\mathsf{fma}\left(0.001736111111111111, x \cdot x, -0.027777777777777776\right) \cdot \left(x \cdot x\right)}{\color{blue}{\mathsf{fma}\left(0.041666666666666664, x, -0.16666666666666666\right)}} \]
10. Step-by-step derivation

    1. lift-+.f64 N/A

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

    2. lift-fma.f64 N/A

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

    3. lift-/.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-fma.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. lift-*.f64 N/A

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

    8. lift-fma.f64 N/A

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

    9. +-commutative N/A

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

11. Applied rewrites 70.3%

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

Alternative 13: 67.6% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.1%

   \[\frac{e^{x} - 1}{x} \]
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

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 70.3

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

5. Applied rewrites 70.3%

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

Alternative 14: 67.3% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.1%

   \[\frac{e^{x} - 1}{x} \]
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

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 70.3

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

5. Applied rewrites 70.3%

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

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24} \cdot x, x, \frac{1}{2}\right), x, 1\right) \]
7. Step-by-step derivation

   1. lower-*.f64 69.7

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

8. Applied rewrites 69.7%

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

Alternative 15: 64.0% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.1%

   \[\frac{e^{x} - 1}{x} \]
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

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 66.3

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

5. Applied rewrites 66.3%

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

Alternative 16: 63.3% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.1%

   \[\frac{e^{x} - 1}{x} \]
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

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 66.3

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

5. Applied rewrites 66.3%

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

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(\frac{1}{6} \cdot x, x, 1\right) \]
7. Step-by-step derivation

   1. lower-*.f64 65.0

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

8. Applied rewrites 65.0%

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

Alternative 17: 51.1% accurate, 16.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.1%

   \[\frac{e^{x} - 1}{x} \]
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

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

   2. lower-fma.f64 51.3

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

5. Applied rewrites 51.3%

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

Alternative 18: 51.0% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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 54.1%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 50.4%

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

Developer Target 1: 52.6% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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 2025037 
(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))