expax (section 3.5)

Percentage Accurate: 53.8% → 100.0%

Time: 2.3s

Alternatives: 5

Speedup: 18.2×

Specification

\[710 > a \cdot x\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, x)
	return Float64(exp(Float64(a * x)) - 1.0)
end

MATLAB

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

Wolfram

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

Initial Program: 53.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(a, x)
	return Float64(exp(Float64(a * x)) - 1.0)
end

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.1%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lower-expm1.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 71.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 1.1

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

   8. Applied rewrites 1.1%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. pow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-/.f64 7.4

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

   11. Applied rewrites 7.4%

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

   12. Taylor expanded in a around 0

       \[\leadsto \left(\frac{a}{x} \cdot x\right) \cdot x - 1 \]
   13. Step-by-step derivation

       1. lift-/.f64 15.1

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

   14. Applied rewrites 15.1%

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

   if -1 < (-.f64 (exp.f64 (*.f64 a x)) #s(literal 1 binary64))

   1. Initial program 32.7%

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

   3. Taylor expanded in a around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   5. Applied rewrites 94.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 98.7

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

   7. Applied rewrites 98.7%

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

Alternative 3: 68.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \cdot x \leq -1 \cdot 10^{+15}:\\ \;\;\;\;\left(\left(\left(a \cdot a\right) \cdot 0.5\right) \cdot x\right) \cdot x - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot x, 0.5 \cdot a, a\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -1e+15)
   (- (* (* (* (* a a) 0.5) x) x) 1.0)
   (* (fma (* a x) (* 0.5 a) a) x)))

C

double code(double a, double x) {
	double tmp;
	if ((a * x) <= -1e+15) {
		tmp = ((((a * a) * 0.5) * x) * x) - 1.0;
	} else {
		tmp = fma((a * x), (0.5 * a), a) * x;
	}
	return tmp;
}

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(a * x) <= -1e+15)
		tmp = Float64(Float64(Float64(Float64(Float64(a * a) * 0.5) * x) * x) - 1.0);
	else
		tmp = Float64(fma(Float64(a * x), Float64(0.5 * a), a) * x);
	end
	return tmp
end

Wolfram

code[a_, x_] := If[LessEqual[N[(a * x), $MachinePrecision], -1e+15], N[(N[(N[(N[(N[(a * a), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(a * x), $MachinePrecision] * N[(0.5 * a), $MachinePrecision] + a), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a x) < -1e15

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 1.1

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

   8. Applied rewrites 1.1%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. pow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-/.f64 7.6

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

   11. Applied rewrites 7.6%

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

   12. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 7.7

          \[\leadsto \left(\left(\left(a \cdot a\right) \cdot 0.5\right) \cdot x\right) \cdot x - 1 \]

   14. Applied rewrites 7.7%

       \[\leadsto \left(\left(\left(a \cdot a\right) \cdot 0.5\right) \cdot x\right) \cdot x - 1 \]

   if -1e15 < (*.f64 a x)

   1. Initial program 33.9%

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

   3. Taylor expanded in a around 0

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

      1. distribute-rgt-in N/A

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

      2. associate-*l* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. distribute-rgt-in N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   5. Applied rewrites 92.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 97.0

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

   7. Applied rewrites 97.0%

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

Alternative 4: 67.3% accurate, 18.2× speedup

Math

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

FPCore

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

C

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

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(a, x)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    code = x * a
end function

Java

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

Python

def code(a, x):
	return x * a

Julia

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

MATLAB

function tmp = code(a, x)
	tmp = x * a;
end

Wolfram

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

Derivation

1. Initial program 56.1%

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

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 65.4

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

5. Applied rewrites 65.4%

   \[\leadsto \color{blue}{x \cdot a} \]
6. Add Preprocessing

Alternative 5: 19.9% accurate, 27.3× speedup

Math

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

FPCore

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

C

double code(double a, double x) {
	return 1.0 - 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(a, x)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    code = 1.0d0 - 1.0d0
end function

Java

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

Python

def code(a, x):
	return 1.0 - 1.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.1%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 20.2%

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

Developer Target 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2025056 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :pre (> 710.0 (* a x))

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

  (- (exp (* a x)) 1.0))