expax (section 3.5)

Percentage Accurate: 54.8% → 99.6%

Time: 2.6s

Alternatives: 10

Speedup: 0.4×

Specification

\[710 > a \cdot x\]

Math

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

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: 54.8% accurate, 1.0× speedup

Math

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

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: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := -0.5 \cdot \left(a \cdot x\right)\\ t_1 := \frac{1}{1 + t\_0}\\ \mathbf{if}\;e^{a \cdot x} - 1 \leq -1:\\ \;\;\;\;t\_1 \cdot t\_1 - 1\\ \mathbf{else}:\\ \;\;\;\;\sinh \left(x \cdot a\right) + \left(\sinh t\_0 \cdot -2\right) \cdot \sinh \left(0.5 \cdot \left(a \cdot x\right)\right)\\ \end{array} \]

FPCore

(FPCore (a x)
  :precision binary64
  (let* ((t_0 (* -0.5 (* a x))) (t_1 (/ 1.0 (+ 1.0 t_0))))
  (if (<= (- (exp (* a x)) 1.0) -1.0)
    (- (* t_1 t_1) 1.0)
    (+
     (sinh (* x a))
     (* (* (sinh t_0) -2.0) (sinh (* 0.5 (* a x))))))))

C

double code(double a, double x) {
	double t_0 = -0.5 * (a * x);
	double t_1 = 1.0 / (1.0 + t_0);
	double tmp;
	if ((exp((a * x)) - 1.0) <= -1.0) {
		tmp = (t_1 * t_1) - 1.0;
	} else {
		tmp = sinh((x * a)) + ((sinh(t_0) * -2.0) * sinh((0.5 * (a * 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(a, x)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-0.5d0) * (a * x)
    t_1 = 1.0d0 / (1.0d0 + t_0)
    if ((exp((a * x)) - 1.0d0) <= (-1.0d0)) then
        tmp = (t_1 * t_1) - 1.0d0
    else
        tmp = sinh((x * a)) + ((sinh(t_0) * (-2.0d0)) * sinh((0.5d0 * (a * x))))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double t_0 = -0.5 * (a * x);
	double t_1 = 1.0 / (1.0 + t_0);
	double tmp;
	if ((Math.exp((a * x)) - 1.0) <= -1.0) {
		tmp = (t_1 * t_1) - 1.0;
	} else {
		tmp = Math.sinh((x * a)) + ((Math.sinh(t_0) * -2.0) * Math.sinh((0.5 * (a * x))));
	}
	return tmp;
}

Python

def code(a, x):
	t_0 = -0.5 * (a * x)
	t_1 = 1.0 / (1.0 + t_0)
	tmp = 0
	if (math.exp((a * x)) - 1.0) <= -1.0:
		tmp = (t_1 * t_1) - 1.0
	else:
		tmp = math.sinh((x * a)) + ((math.sinh(t_0) * -2.0) * math.sinh((0.5 * (a * x))))
	return tmp

Julia

function code(a, x)
	t_0 = Float64(-0.5 * Float64(a * x))
	t_1 = Float64(1.0 / Float64(1.0 + t_0))
	tmp = 0.0
	if (Float64(exp(Float64(a * x)) - 1.0) <= -1.0)
		tmp = Float64(Float64(t_1 * t_1) - 1.0);
	else
		tmp = Float64(sinh(Float64(x * a)) + Float64(Float64(sinh(t_0) * -2.0) * sinh(Float64(0.5 * Float64(a * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	t_0 = -0.5 * (a * x);
	t_1 = 1.0 / (1.0 + t_0);
	tmp = 0.0;
	if ((exp((a * x)) - 1.0) <= -1.0)
		tmp = (t_1 * t_1) - 1.0;
	else
		tmp = sinh((x * a)) + ((sinh(t_0) * -2.0) * sinh((0.5 * (a * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := Block[{t$95$0 = N[(-0.5 * N[(a * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Exp[N[(a * x), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision], -1.0], N[(N[(t$95$1 * t$95$1), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[Sinh[N[(x * a), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Sinh[t$95$0], $MachinePrecision] * -2.0), $MachinePrecision] * N[Sinh[N[(0.5 * N[(a * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 54.8%

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

      1. lift-exp.f64 N/A

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

      2. exp-fabs N/A

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

      3. lift-exp.f64 N/A

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

      4. rem-sqrt-square-rev N/A

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

      5. sqrt-prod N/A

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

      6. lower-unsound-*.f64 N/A

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

      7. lower-unsound-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-unsound-sqrt.f64 54.8%

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 54.8%

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

   3. Applied rewrites 54.8%

      \[\leadsto \color{blue}{\sqrt{e^{x \cdot a}} \cdot \sqrt{e^{x \cdot a}}} - 1 \]
   4. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. exp-sqrt-rev N/A

         \[\leadsto \color{blue}{e^{\frac{x \cdot a}{2}}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      4. +-lft-identity N/A

         \[\leadsto e^{\frac{\color{blue}{0 + x \cdot a}}{2}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      5. lift-*.f64 N/A

         \[\leadsto e^{\frac{0 + \color{blue}{x \cdot a}}{2}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

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

         \[\leadsto e^{\frac{\color{blue}{0 - \left(\mathsf{neg}\left(x\right)\right) \cdot a}}{2}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      7. lift-neg.f64 N/A

         \[\leadsto e^{\frac{0 - \color{blue}{\left(-x\right)} \cdot a}{2}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      8. lift-*.f64 N/A

         \[\leadsto e^{\frac{0 - \color{blue}{\left(-x\right) \cdot a}}{2}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      9. div-sub N/A

         \[\leadsto e^{\color{blue}{\frac{0}{2} - \frac{\left(-x\right) \cdot a}{2}}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      10. metadata-eval N/A

          \[\leadsto e^{\color{blue}{0} - \frac{\left(-x\right) \cdot a}{2}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      11. mult-flip-rev N/A

          \[\leadsto e^{0 - \color{blue}{\left(\left(-x\right) \cdot a\right) \cdot \frac{1}{2}}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      12. lift-*.f64 N/A

          \[\leadsto e^{0 - \color{blue}{\left(\left(-x\right) \cdot a\right)} \cdot \frac{1}{2}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      13. lift-neg.f64 N/A

          \[\leadsto e^{0 - \left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot a\right) \cdot \frac{1}{2}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      14. distribute-lft-neg-out N/A

          \[\leadsto e^{0 - \color{blue}{\left(\mathsf{neg}\left(x \cdot a\right)\right)} \cdot \frac{1}{2}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      15. lift-*.f64 N/A

          \[\leadsto e^{0 - \left(\mathsf{neg}\left(\color{blue}{x \cdot a}\right)\right) \cdot \frac{1}{2}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      16. sub0-neg N/A

          \[\leadsto e^{0 - \color{blue}{\left(0 - x \cdot a\right)} \cdot \frac{1}{2}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      17. lift--.f64 N/A

          \[\leadsto e^{0 - \color{blue}{\left(0 - x \cdot a\right)} \cdot \frac{1}{2}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      18. metadata-eval N/A

          \[\leadsto e^{0 - \left(0 - x \cdot a\right) \cdot \color{blue}{\frac{1}{2}}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      19. lift-*.f64 N/A

          \[\leadsto e^{0 - \color{blue}{\left(0 - x \cdot a\right) \cdot \frac{1}{2}}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      20. div-exp N/A

          \[\leadsto \color{blue}{\frac{e^{0}}{e^{\left(0 - x \cdot a\right) \cdot \frac{1}{2}}}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      21. 1-exp N/A

          \[\leadsto \frac{\color{blue}{1}}{e^{\left(0 - x \cdot a\right) \cdot \frac{1}{2}}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      22. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{e^{\left(0 - x \cdot a\right) \cdot \frac{1}{2}}}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

   5. Applied rewrites 54.8%

      \[\leadsto \color{blue}{\frac{1}{e^{-0.5 \cdot \left(a \cdot x\right)}}} \cdot \sqrt{e^{x \cdot a}} - 1 \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. exp-sqrt-rev N/A

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

      4. +-lft-identity N/A

         \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{\frac{\color{blue}{0 + x \cdot a}}{2}} - 1 \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{\frac{0 + \color{blue}{x \cdot a}}{2}} - 1 \]

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

         \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{\frac{\color{blue}{0 - \left(\mathsf{neg}\left(x\right)\right) \cdot a}}{2}} - 1 \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{\frac{0 - \color{blue}{\left(-x\right)} \cdot a}{2}} - 1 \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{\frac{0 - \color{blue}{\left(-x\right) \cdot a}}{2}} - 1 \]

      9. div-sub N/A

         \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{\color{blue}{\frac{0}{2} - \frac{\left(-x\right) \cdot a}{2}}} - 1 \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{\color{blue}{0} - \frac{\left(-x\right) \cdot a}{2}} - 1 \]

      11. mult-flip-rev N/A

          \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{0 - \color{blue}{\left(\left(-x\right) \cdot a\right) \cdot \frac{1}{2}}} - 1 \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{0 - \color{blue}{\left(\left(-x\right) \cdot a\right)} \cdot \frac{1}{2}} - 1 \]

      13. lift-neg.f64 N/A

          \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{0 - \left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot a\right) \cdot \frac{1}{2}} - 1 \]

      14. distribute-lft-neg-out N/A

          \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{0 - \color{blue}{\left(\mathsf{neg}\left(x \cdot a\right)\right)} \cdot \frac{1}{2}} - 1 \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{0 - \left(\mathsf{neg}\left(\color{blue}{x \cdot a}\right)\right) \cdot \frac{1}{2}} - 1 \]

      16. sub0-neg N/A

          \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{0 - \color{blue}{\left(0 - x \cdot a\right)} \cdot \frac{1}{2}} - 1 \]

      17. lift--.f64 N/A

          \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{0 - \color{blue}{\left(0 - x \cdot a\right)} \cdot \frac{1}{2}} - 1 \]

      18. metadata-eval N/A

          \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{0 - \left(0 - x \cdot a\right) \cdot \color{blue}{\frac{1}{2}}} - 1 \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{0 - \color{blue}{\left(0 - x \cdot a\right) \cdot \frac{1}{2}}} - 1 \]

      20. div-exp N/A

          \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot \color{blue}{\frac{e^{0}}{e^{\left(0 - x \cdot a\right) \cdot \frac{1}{2}}}} - 1 \]

      21. 1-exp N/A

          \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot \frac{\color{blue}{1}}{e^{\left(0 - x \cdot a\right) \cdot \frac{1}{2}}} - 1 \]

      22. lower-/.f64 N/A

          \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot \color{blue}{\frac{1}{e^{\left(0 - x \cdot a\right) \cdot \frac{1}{2}}}} - 1 \]

   7. Applied rewrites 54.8%

      \[\leadsto \frac{1}{e^{-0.5 \cdot \left(a \cdot x\right)}} \cdot \color{blue}{\frac{1}{e^{-0.5 \cdot \left(a \cdot x\right)}}} - 1 \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{1}{\color{blue}{1 + \frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot \frac{1}{e^{-0.5 \cdot \left(a \cdot x\right)}} - 1 \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 53.6%

         \[\leadsto \frac{1}{1 + -0.5 \cdot \left(a \cdot \color{blue}{x}\right)} \cdot \frac{1}{e^{-0.5 \cdot \left(a \cdot x\right)}} - 1 \]

   10. Applied rewrites 53.6%

       \[\leadsto \frac{1}{\color{blue}{1 + -0.5 \cdot \left(a \cdot x\right)}} \cdot \frac{1}{e^{-0.5 \cdot \left(a \cdot x\right)}} - 1 \]

   11. Taylor expanded in a around 0

       \[\leadsto \frac{1}{1 + -0.5 \cdot \left(a \cdot x\right)} \cdot \frac{1}{\color{blue}{1 + \frac{-1}{2} \cdot \left(a \cdot x\right)}} - 1 \]
   12. Step-by-step derivation

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 53.3%

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

   13. Applied rewrites 53.3%

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

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

   1. Initial program 54.8%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(1 - e^{a \cdot x}\right)\right)} \]

      3. lift-exp.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(1 - \color{blue}{e^{a \cdot x}}\right)\right) \]

      4. sinh-+-cosh-rev N/A

         \[\leadsto \mathsf{neg}\left(\left(1 - \color{blue}{\left(\cosh \left(a \cdot x\right) + \sinh \left(a \cdot x\right)\right)}\right)\right) \]

      5. associate--r+ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\left(1 - \cosh \left(a \cdot x\right)\right) - \sinh \left(a \cdot x\right)\right)}\right) \]

      6. sub-negate N/A

         \[\leadsto \color{blue}{\sinh \left(a \cdot x\right) - \left(1 - \cosh \left(a \cdot x\right)\right)} \]

      7. cosh-0-rev N/A

         \[\leadsto \sinh \left(a \cdot x\right) - \left(\color{blue}{\cosh 0} - \cosh \left(a \cdot x\right)\right) \]

      8. cosh-neg-rev N/A

         \[\leadsto \sinh \left(a \cdot x\right) - \left(\cosh 0 - \color{blue}{\cosh \left(\mathsf{neg}\left(a \cdot x\right)\right)}\right) \]

      9. diff-cosh N/A

         \[\leadsto \sinh \left(a \cdot x\right) - \color{blue}{2 \cdot \left(\sinh \left(\frac{0 + \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right) \cdot \sinh \left(\frac{0 - \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right)\right)} \]

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

          \[\leadsto \color{blue}{\sinh \left(a \cdot x\right) + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\sinh \left(\frac{0 + \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right) \cdot \sinh \left(\frac{0 - \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right)\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \color{blue}{\sinh \left(a \cdot x\right) + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\sinh \left(\frac{0 + \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right) \cdot \sinh \left(\frac{0 - \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right)\right)} \]

      12. lower-sinh.f64 N/A

          \[\leadsto \color{blue}{\sinh \left(a \cdot x\right)} + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\sinh \left(\frac{0 + \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right) \cdot \sinh \left(\frac{0 - \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sinh \color{blue}{\left(a \cdot x\right)} + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\sinh \left(\frac{0 + \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right) \cdot \sinh \left(\frac{0 - \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \sinh \color{blue}{\left(x \cdot a\right)} + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\sinh \left(\frac{0 + \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right) \cdot \sinh \left(\frac{0 - \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \sinh \color{blue}{\left(x \cdot a\right)} + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\sinh \left(\frac{0 + \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right) \cdot \sinh \left(\frac{0 - \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \sinh \left(x \cdot a\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \left(\sinh \left(\frac{0 + \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right) \cdot \sinh \left(\frac{0 - \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right)\right)} \]

      17. metadata-eval N/A

          \[\leadsto \sinh \left(x \cdot a\right) + \color{blue}{-2} \cdot \left(\sinh \left(\frac{0 + \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right) \cdot \sinh \left(\frac{0 - \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right)\right) \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\sinh \left(x \cdot a\right) + -2 \cdot \left(\sinh \left(\left(0 - x \cdot a\right) \cdot 0.5\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot 0.5\right)\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sinh \left(x \cdot a\right) + \color{blue}{-2 \cdot \left(\sinh \left(\left(0 - x \cdot a\right) \cdot \frac{1}{2}\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sinh \left(x \cdot a\right) + -2 \cdot \color{blue}{\left(\sinh \left(\left(0 - x \cdot a\right) \cdot \frac{1}{2}\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sinh \left(x \cdot a\right) + \color{blue}{\left(-2 \cdot \sinh \left(\left(0 - x \cdot a\right) \cdot \frac{1}{2}\right)\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sinh \left(x \cdot a\right) + \color{blue}{\left(-2 \cdot \sinh \left(\left(0 - x \cdot a\right) \cdot \frac{1}{2}\right)\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right)} \]

      5. *-commutative N/A

         \[\leadsto \sinh \left(x \cdot a\right) + \color{blue}{\left(\sinh \left(\left(0 - x \cdot a\right) \cdot \frac{1}{2}\right) \cdot -2\right)} \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right) \]

      6. lower-*.f64 66.2%

         \[\leadsto \sinh \left(x \cdot a\right) + \color{blue}{\left(\sinh \left(\left(0 - x \cdot a\right) \cdot 0.5\right) \cdot -2\right)} \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot 0.5\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \sinh \left(x \cdot a\right) + \left(\sinh \color{blue}{\left(\left(0 - x \cdot a\right) \cdot \frac{1}{2}\right)} \cdot -2\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right) \]

      8. lift--.f64 N/A

         \[\leadsto \sinh \left(x \cdot a\right) + \left(\sinh \left(\color{blue}{\left(0 - x \cdot a\right)} \cdot \frac{1}{2}\right) \cdot -2\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right) \]

      9. sub0-neg N/A

         \[\leadsto \sinh \left(x \cdot a\right) + \left(\sinh \left(\color{blue}{\left(\mathsf{neg}\left(x \cdot a\right)\right)} \cdot \frac{1}{2}\right) \cdot -2\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right) \]

      10. distribute-lft-neg-out N/A

          \[\leadsto \sinh \left(x \cdot a\right) + \left(\sinh \color{blue}{\left(\mathsf{neg}\left(\left(x \cdot a\right) \cdot \frac{1}{2}\right)\right)} \cdot -2\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right) \]

      11. *-commutative N/A

          \[\leadsto \sinh \left(x \cdot a\right) + \left(\sinh \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \left(x \cdot a\right)}\right)\right) \cdot -2\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right) \]

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

          \[\leadsto \sinh \left(x \cdot a\right) + \left(\sinh \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left(x \cdot a\right)\right)} \cdot -2\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \sinh \left(x \cdot a\right) + \left(\sinh \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left(x \cdot a\right)\right)} \cdot -2\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right) \]

      14. metadata-eval 66.2%

          \[\leadsto \sinh \left(x \cdot a\right) + \left(\sinh \left(\color{blue}{-0.5} \cdot \left(x \cdot a\right)\right) \cdot -2\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot 0.5\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \sinh \left(x \cdot a\right) + \left(\sinh \left(\frac{-1}{2} \cdot \color{blue}{\left(x \cdot a\right)}\right) \cdot -2\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right) \]

      16. *-commutative N/A

          \[\leadsto \sinh \left(x \cdot a\right) + \left(\sinh \left(\frac{-1}{2} \cdot \color{blue}{\left(a \cdot x\right)}\right) \cdot -2\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right) \]

      17. lower-*.f64 66.2%

          \[\leadsto \sinh \left(x \cdot a\right) + \left(\sinh \left(-0.5 \cdot \color{blue}{\left(a \cdot x\right)}\right) \cdot -2\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot 0.5\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \sinh \left(x \cdot a\right) + \left(\sinh \left(\frac{-1}{2} \cdot \left(a \cdot x\right)\right) \cdot -2\right) \cdot \sinh \color{blue}{\left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right)} \]

   5. Applied rewrites 66.2%

      \[\leadsto \sinh \left(x \cdot a\right) + \color{blue}{\left(\sinh \left(-0.5 \cdot \left(a \cdot x\right)\right) \cdot -2\right) \cdot \sinh \left(0.5 \cdot \left(a \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;e^{a \cdot x} - 1 \leq -0.0001:\\ \;\;\;\;\frac{-1 \cdot -1 - e^{x \cdot \left(a + a\right)}}{-1 - e^{x \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\sinh \left(x \cdot a\right) - \left(1 - \cosh \left(x \cdot a\right)\right)\\ \end{array} \]

FPCore

(FPCore (a x)
  :precision binary64
  (if (<= (- (exp (* a x)) 1.0) -0.0001)
  (/ (- (* -1.0 -1.0) (exp (* x (+ a a)))) (- -1.0 (exp (* x a))))
  (- (sinh (* x a)) (- 1.0 (cosh (* x a))))))

C

double code(double a, double x) {
	double tmp;
	if ((exp((a * x)) - 1.0) <= -0.0001) {
		tmp = ((-1.0 * -1.0) - exp((x * (a + a)))) / (-1.0 - exp((x * a)));
	} else {
		tmp = sinh((x * a)) - (1.0 - cosh((x * a)));
	}
	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(a, x)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((exp((a * x)) - 1.0d0) <= (-0.0001d0)) then
        tmp = (((-1.0d0) * (-1.0d0)) - exp((x * (a + a)))) / ((-1.0d0) - exp((x * a)))
    else
        tmp = sinh((x * a)) - (1.0d0 - cosh((x * a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double tmp;
	if ((Math.exp((a * x)) - 1.0) <= -0.0001) {
		tmp = ((-1.0 * -1.0) - Math.exp((x * (a + a)))) / (-1.0 - Math.exp((x * a)));
	} else {
		tmp = Math.sinh((x * a)) - (1.0 - Math.cosh((x * a)));
	}
	return tmp;
}

Python

def code(a, x):
	tmp = 0
	if (math.exp((a * x)) - 1.0) <= -0.0001:
		tmp = ((-1.0 * -1.0) - math.exp((x * (a + a)))) / (-1.0 - math.exp((x * a)))
	else:
		tmp = math.sinh((x * a)) - (1.0 - math.cosh((x * a)))
	return tmp

Julia

function code(a, x)
	tmp = 0.0
	if (Float64(exp(Float64(a * x)) - 1.0) <= -0.0001)
		tmp = Float64(Float64(Float64(-1.0 * -1.0) - exp(Float64(x * Float64(a + a)))) / Float64(-1.0 - exp(Float64(x * a))));
	else
		tmp = Float64(sinh(Float64(x * a)) - Float64(1.0 - cosh(Float64(x * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	tmp = 0.0;
	if ((exp((a * x)) - 1.0) <= -0.0001)
		tmp = ((-1.0 * -1.0) - exp((x * (a + a)))) / (-1.0 - exp((x * a)));
	else
		tmp = sinh((x * a)) - (1.0 - cosh((x * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := If[LessEqual[N[(N[Exp[N[(a * x), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision], -0.0001], N[(N[(N[(-1.0 * -1.0), $MachinePrecision] - N[Exp[N[(x * N[(a + a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[Exp[N[(x * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sinh[N[(x * a), $MachinePrecision]], $MachinePrecision] - N[(1.0 - N[Cosh[N[(x * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (*.f64 a x)) #s(literal 1 binary64)) < -1e-4

   1. Initial program 54.8%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

         \[\leadsto \color{blue}{e^{a \cdot x} + \left(\mathsf{neg}\left(1\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(1\right)\right) + e^{a \cdot x}} \]

      4. flip-+ N/A

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

      5. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 54.8%

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

   if -1e-4 < (-.f64 (exp.f64 (*.f64 a x)) #s(literal 1 binary64))

   1. Initial program 54.8%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(1 - e^{a \cdot x}\right)\right)} \]

      3. lift-exp.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(1 - \color{blue}{e^{a \cdot x}}\right)\right) \]

      4. sinh-+-cosh-rev N/A

         \[\leadsto \mathsf{neg}\left(\left(1 - \color{blue}{\left(\cosh \left(a \cdot x\right) + \sinh \left(a \cdot x\right)\right)}\right)\right) \]

      5. associate--r+ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\left(1 - \cosh \left(a \cdot x\right)\right) - \sinh \left(a \cdot x\right)\right)}\right) \]

      6. sub-negate N/A

         \[\leadsto \color{blue}{\sinh \left(a \cdot x\right) - \left(1 - \cosh \left(a \cdot x\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\sinh \left(a \cdot x\right) - \left(1 - \cosh \left(a \cdot x\right)\right)} \]

      8. lower-sinh.f64 N/A

         \[\leadsto \color{blue}{\sinh \left(a \cdot x\right)} - \left(1 - \cosh \left(a \cdot x\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \sinh \color{blue}{\left(a \cdot x\right)} - \left(1 - \cosh \left(a \cdot x\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \sinh \color{blue}{\left(x \cdot a\right)} - \left(1 - \cosh \left(a \cdot x\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \sinh \color{blue}{\left(x \cdot a\right)} - \left(1 - \cosh \left(a \cdot x\right)\right) \]

      12. lower--.f64 N/A

          \[\leadsto \sinh \left(x \cdot a\right) - \color{blue}{\left(1 - \cosh \left(a \cdot x\right)\right)} \]

      13. lower-cosh.f64 65.6%

          \[\leadsto \sinh \left(x \cdot a\right) - \left(1 - \color{blue}{\cosh \left(a \cdot x\right)}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sinh \left(x \cdot a\right) - \left(1 - \cosh \color{blue}{\left(a \cdot x\right)}\right) \]

      15. *-commutative N/A

          \[\leadsto \sinh \left(x \cdot a\right) - \left(1 - \cosh \color{blue}{\left(x \cdot a\right)}\right) \]

      16. lower-*.f64 65.6%

          \[\leadsto \sinh \left(x \cdot a\right) - \left(1 - \cosh \color{blue}{\left(x \cdot a\right)}\right) \]

   3. Applied rewrites 65.6%

      \[\leadsto \color{blue}{\sinh \left(x \cdot a\right) - \left(1 - \cosh \left(x \cdot a\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := e^{a \cdot x} - 1\\ \mathbf{if}\;t\_0 \leq -0.0001:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sinh \left(x \cdot a\right) - \left(1 - \cosh \left(x \cdot a\right)\right)\\ \end{array} \]

FPCore

(FPCore (a x)
  :precision binary64
  (let* ((t_0 (- (exp (* a x)) 1.0)))
  (if (<= t_0 -0.0001) t_0 (- (sinh (* x a)) (- 1.0 (cosh (* x a)))))))

C

double code(double a, double x) {
	double t_0 = exp((a * x)) - 1.0;
	double tmp;
	if (t_0 <= -0.0001) {
		tmp = t_0;
	} else {
		tmp = sinh((x * a)) - (1.0 - cosh((x * a)));
	}
	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(a, x)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((a * x)) - 1.0d0
    if (t_0 <= (-0.0001d0)) then
        tmp = t_0
    else
        tmp = sinh((x * a)) - (1.0d0 - cosh((x * a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double t_0 = Math.exp((a * x)) - 1.0;
	double tmp;
	if (t_0 <= -0.0001) {
		tmp = t_0;
	} else {
		tmp = Math.sinh((x * a)) - (1.0 - Math.cosh((x * a)));
	}
	return tmp;
}

Python

def code(a, x):
	t_0 = math.exp((a * x)) - 1.0
	tmp = 0
	if t_0 <= -0.0001:
		tmp = t_0
	else:
		tmp = math.sinh((x * a)) - (1.0 - math.cosh((x * a)))
	return tmp

Julia

function code(a, x)
	t_0 = Float64(exp(Float64(a * x)) - 1.0)
	tmp = 0.0
	if (t_0 <= -0.0001)
		tmp = t_0;
	else
		tmp = Float64(sinh(Float64(x * a)) - Float64(1.0 - cosh(Float64(x * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	t_0 = exp((a * x)) - 1.0;
	tmp = 0.0;
	if (t_0 <= -0.0001)
		tmp = t_0;
	else
		tmp = sinh((x * a)) - (1.0 - cosh((x * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := Block[{t$95$0 = N[(N[Exp[N[(a * x), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -0.0001], t$95$0, N[(N[Sinh[N[(x * a), $MachinePrecision]], $MachinePrecision] - N[(1.0 - N[Cosh[N[(x * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (*.f64 a x)) #s(literal 1 binary64)) < -1e-4

   1. Initial program 54.8%

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

   if -1e-4 < (-.f64 (exp.f64 (*.f64 a x)) #s(literal 1 binary64))

   1. Initial program 54.8%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(1 - e^{a \cdot x}\right)\right)} \]

      3. lift-exp.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(1 - \color{blue}{e^{a \cdot x}}\right)\right) \]

      4. sinh-+-cosh-rev N/A

         \[\leadsto \mathsf{neg}\left(\left(1 - \color{blue}{\left(\cosh \left(a \cdot x\right) + \sinh \left(a \cdot x\right)\right)}\right)\right) \]

      5. associate--r+ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\left(1 - \cosh \left(a \cdot x\right)\right) - \sinh \left(a \cdot x\right)\right)}\right) \]

      6. sub-negate N/A

         \[\leadsto \color{blue}{\sinh \left(a \cdot x\right) - \left(1 - \cosh \left(a \cdot x\right)\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\sinh \left(a \cdot x\right) - \left(1 - \cosh \left(a \cdot x\right)\right)} \]

      8. lower-sinh.f64 N/A

         \[\leadsto \color{blue}{\sinh \left(a \cdot x\right)} - \left(1 - \cosh \left(a \cdot x\right)\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \sinh \color{blue}{\left(a \cdot x\right)} - \left(1 - \cosh \left(a \cdot x\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \sinh \color{blue}{\left(x \cdot a\right)} - \left(1 - \cosh \left(a \cdot x\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \sinh \color{blue}{\left(x \cdot a\right)} - \left(1 - \cosh \left(a \cdot x\right)\right) \]

      12. lower--.f64 N/A

          \[\leadsto \sinh \left(x \cdot a\right) - \color{blue}{\left(1 - \cosh \left(a \cdot x\right)\right)} \]

      13. lower-cosh.f64 65.6%

          \[\leadsto \sinh \left(x \cdot a\right) - \left(1 - \color{blue}{\cosh \left(a \cdot x\right)}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sinh \left(x \cdot a\right) - \left(1 - \cosh \color{blue}{\left(a \cdot x\right)}\right) \]

      15. *-commutative N/A

          \[\leadsto \sinh \left(x \cdot a\right) - \left(1 - \cosh \color{blue}{\left(x \cdot a\right)}\right) \]

      16. lower-*.f64 65.6%

          \[\leadsto \sinh \left(x \cdot a\right) - \left(1 - \cosh \color{blue}{\left(x \cdot a\right)}\right) \]

   3. Applied rewrites 65.6%

      \[\leadsto \color{blue}{\sinh \left(x \cdot a\right) - \left(1 - \cosh \left(x \cdot a\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := e^{a \cdot x} - 1\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sinh \left(x \cdot a\right) + \left(-1 \cdot \left(x \cdot \left(-0.5 \cdot a - 0.5 \cdot a\right)\right)\right) \cdot \left(0.5 \cdot \left(x \cdot \left(0.5 \cdot a - -0.5 \cdot a\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a x)
  :precision binary64
  (let* ((t_0 (- (exp (* a x)) 1.0)))
  (if (<= t_0 -0.02)
    t_0
    (+
     (sinh (* x a))
     (*
      (* -1.0 (* x (- (* -0.5 a) (* 0.5 a))))
      (* 0.5 (* x (- (* 0.5 a) (* -0.5 a)))))))))

C

double code(double a, double x) {
	double t_0 = exp((a * x)) - 1.0;
	double tmp;
	if (t_0 <= -0.02) {
		tmp = t_0;
	} else {
		tmp = sinh((x * a)) + ((-1.0 * (x * ((-0.5 * a) - (0.5 * a)))) * (0.5 * (x * ((0.5 * a) - (-0.5 * a)))));
	}
	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(a, x)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((a * x)) - 1.0d0
    if (t_0 <= (-0.02d0)) then
        tmp = t_0
    else
        tmp = sinh((x * a)) + (((-1.0d0) * (x * (((-0.5d0) * a) - (0.5d0 * a)))) * (0.5d0 * (x * ((0.5d0 * a) - ((-0.5d0) * a)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double t_0 = Math.exp((a * x)) - 1.0;
	double tmp;
	if (t_0 <= -0.02) {
		tmp = t_0;
	} else {
		tmp = Math.sinh((x * a)) + ((-1.0 * (x * ((-0.5 * a) - (0.5 * a)))) * (0.5 * (x * ((0.5 * a) - (-0.5 * a)))));
	}
	return tmp;
}

Python

def code(a, x):
	t_0 = math.exp((a * x)) - 1.0
	tmp = 0
	if t_0 <= -0.02:
		tmp = t_0
	else:
		tmp = math.sinh((x * a)) + ((-1.0 * (x * ((-0.5 * a) - (0.5 * a)))) * (0.5 * (x * ((0.5 * a) - (-0.5 * a)))))
	return tmp

Julia

function code(a, x)
	t_0 = Float64(exp(Float64(a * x)) - 1.0)
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = t_0;
	else
		tmp = Float64(sinh(Float64(x * a)) + Float64(Float64(-1.0 * Float64(x * Float64(Float64(-0.5 * a) - Float64(0.5 * a)))) * Float64(0.5 * Float64(x * Float64(Float64(0.5 * a) - Float64(-0.5 * a))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	t_0 = exp((a * x)) - 1.0;
	tmp = 0.0;
	if (t_0 <= -0.02)
		tmp = t_0;
	else
		tmp = sinh((x * a)) + ((-1.0 * (x * ((-0.5 * a) - (0.5 * a)))) * (0.5 * (x * ((0.5 * a) - (-0.5 * a)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := Block[{t$95$0 = N[(N[Exp[N[(a * x), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], t$95$0, N[(N[Sinh[N[(x * a), $MachinePrecision]], $MachinePrecision] + N[(N[(-1.0 * N[(x * N[(N[(-0.5 * a), $MachinePrecision] - N[(0.5 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(x * N[(N[(0.5 * a), $MachinePrecision] - N[(-0.5 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (*.f64 a x)) #s(literal 1 binary64)) < -0.02

   1. Initial program 54.8%

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

   if -0.02 < (-.f64 (exp.f64 (*.f64 a x)) #s(literal 1 binary64))

   1. Initial program 54.8%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(1 - e^{a \cdot x}\right)\right)} \]

      3. lift-exp.f64 N/A

         \[\leadsto \mathsf{neg}\left(\left(1 - \color{blue}{e^{a \cdot x}}\right)\right) \]

      4. sinh-+-cosh-rev N/A

         \[\leadsto \mathsf{neg}\left(\left(1 - \color{blue}{\left(\cosh \left(a \cdot x\right) + \sinh \left(a \cdot x\right)\right)}\right)\right) \]

      5. associate--r+ N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\left(1 - \cosh \left(a \cdot x\right)\right) - \sinh \left(a \cdot x\right)\right)}\right) \]

      6. sub-negate N/A

         \[\leadsto \color{blue}{\sinh \left(a \cdot x\right) - \left(1 - \cosh \left(a \cdot x\right)\right)} \]

      7. cosh-0-rev N/A

         \[\leadsto \sinh \left(a \cdot x\right) - \left(\color{blue}{\cosh 0} - \cosh \left(a \cdot x\right)\right) \]

      8. cosh-neg-rev N/A

         \[\leadsto \sinh \left(a \cdot x\right) - \left(\cosh 0 - \color{blue}{\cosh \left(\mathsf{neg}\left(a \cdot x\right)\right)}\right) \]

      9. diff-cosh N/A

         \[\leadsto \sinh \left(a \cdot x\right) - \color{blue}{2 \cdot \left(\sinh \left(\frac{0 + \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right) \cdot \sinh \left(\frac{0 - \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right)\right)} \]

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

          \[\leadsto \color{blue}{\sinh \left(a \cdot x\right) + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\sinh \left(\frac{0 + \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right) \cdot \sinh \left(\frac{0 - \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right)\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \color{blue}{\sinh \left(a \cdot x\right) + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\sinh \left(\frac{0 + \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right) \cdot \sinh \left(\frac{0 - \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right)\right)} \]

      12. lower-sinh.f64 N/A

          \[\leadsto \color{blue}{\sinh \left(a \cdot x\right)} + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\sinh \left(\frac{0 + \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right) \cdot \sinh \left(\frac{0 - \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \sinh \color{blue}{\left(a \cdot x\right)} + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\sinh \left(\frac{0 + \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right) \cdot \sinh \left(\frac{0 - \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \sinh \color{blue}{\left(x \cdot a\right)} + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\sinh \left(\frac{0 + \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right) \cdot \sinh \left(\frac{0 - \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right)\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \sinh \color{blue}{\left(x \cdot a\right)} + \left(\mathsf{neg}\left(2\right)\right) \cdot \left(\sinh \left(\frac{0 + \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right) \cdot \sinh \left(\frac{0 - \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \sinh \left(x \cdot a\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \left(\sinh \left(\frac{0 + \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right) \cdot \sinh \left(\frac{0 - \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right)\right)} \]

      17. metadata-eval N/A

          \[\leadsto \sinh \left(x \cdot a\right) + \color{blue}{-2} \cdot \left(\sinh \left(\frac{0 + \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right) \cdot \sinh \left(\frac{0 - \left(\mathsf{neg}\left(a \cdot x\right)\right)}{2}\right)\right) \]

   3. Applied rewrites 66.2%

      \[\leadsto \color{blue}{\sinh \left(x \cdot a\right) + -2 \cdot \left(\sinh \left(\left(0 - x \cdot a\right) \cdot 0.5\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot 0.5\right)\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sinh \left(x \cdot a\right) + \color{blue}{-2 \cdot \left(\sinh \left(\left(0 - x \cdot a\right) \cdot \frac{1}{2}\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sinh \left(x \cdot a\right) + -2 \cdot \color{blue}{\left(\sinh \left(\left(0 - x \cdot a\right) \cdot \frac{1}{2}\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right)\right)} \]

      3. associate-*r* N/A

         \[\leadsto \sinh \left(x \cdot a\right) + \color{blue}{\left(-2 \cdot \sinh \left(\left(0 - x \cdot a\right) \cdot \frac{1}{2}\right)\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sinh \left(x \cdot a\right) + \color{blue}{\left(-2 \cdot \sinh \left(\left(0 - x \cdot a\right) \cdot \frac{1}{2}\right)\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right)} \]

      5. *-commutative N/A

         \[\leadsto \sinh \left(x \cdot a\right) + \color{blue}{\left(\sinh \left(\left(0 - x \cdot a\right) \cdot \frac{1}{2}\right) \cdot -2\right)} \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right) \]

      6. lower-*.f64 66.2%

         \[\leadsto \sinh \left(x \cdot a\right) + \color{blue}{\left(\sinh \left(\left(0 - x \cdot a\right) \cdot 0.5\right) \cdot -2\right)} \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot 0.5\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \sinh \left(x \cdot a\right) + \left(\sinh \color{blue}{\left(\left(0 - x \cdot a\right) \cdot \frac{1}{2}\right)} \cdot -2\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right) \]

      8. lift--.f64 N/A

         \[\leadsto \sinh \left(x \cdot a\right) + \left(\sinh \left(\color{blue}{\left(0 - x \cdot a\right)} \cdot \frac{1}{2}\right) \cdot -2\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right) \]

      9. sub0-neg N/A

         \[\leadsto \sinh \left(x \cdot a\right) + \left(\sinh \left(\color{blue}{\left(\mathsf{neg}\left(x \cdot a\right)\right)} \cdot \frac{1}{2}\right) \cdot -2\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right) \]

      10. distribute-lft-neg-out N/A

          \[\leadsto \sinh \left(x \cdot a\right) + \left(\sinh \color{blue}{\left(\mathsf{neg}\left(\left(x \cdot a\right) \cdot \frac{1}{2}\right)\right)} \cdot -2\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right) \]

      11. *-commutative N/A

          \[\leadsto \sinh \left(x \cdot a\right) + \left(\sinh \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \left(x \cdot a\right)}\right)\right) \cdot -2\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right) \]

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

          \[\leadsto \sinh \left(x \cdot a\right) + \left(\sinh \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left(x \cdot a\right)\right)} \cdot -2\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \sinh \left(x \cdot a\right) + \left(\sinh \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left(x \cdot a\right)\right)} \cdot -2\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right) \]

      14. metadata-eval 66.2%

          \[\leadsto \sinh \left(x \cdot a\right) + \left(\sinh \left(\color{blue}{-0.5} \cdot \left(x \cdot a\right)\right) \cdot -2\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot 0.5\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \sinh \left(x \cdot a\right) + \left(\sinh \left(\frac{-1}{2} \cdot \color{blue}{\left(x \cdot a\right)}\right) \cdot -2\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right) \]

      16. *-commutative N/A

          \[\leadsto \sinh \left(x \cdot a\right) + \left(\sinh \left(\frac{-1}{2} \cdot \color{blue}{\left(a \cdot x\right)}\right) \cdot -2\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right) \]

      17. lower-*.f64 66.2%

          \[\leadsto \sinh \left(x \cdot a\right) + \left(\sinh \left(-0.5 \cdot \color{blue}{\left(a \cdot x\right)}\right) \cdot -2\right) \cdot \sinh \left(\left(0 + x \cdot a\right) \cdot 0.5\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \sinh \left(x \cdot a\right) + \left(\sinh \left(\frac{-1}{2} \cdot \left(a \cdot x\right)\right) \cdot -2\right) \cdot \sinh \color{blue}{\left(\left(0 + x \cdot a\right) \cdot \frac{1}{2}\right)} \]

   5. Applied rewrites 66.2%

      \[\leadsto \sinh \left(x \cdot a\right) + \color{blue}{\left(\sinh \left(-0.5 \cdot \left(a \cdot x\right)\right) \cdot -2\right) \cdot \sinh \left(0.5 \cdot \left(a \cdot x\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \sinh \left(x \cdot a\right) + \color{blue}{\left(-1 \cdot \left(x \cdot \left(\frac{-1}{2} \cdot a - \frac{1}{2} \cdot a\right)\right)\right)} \cdot \sinh \left(0.5 \cdot \left(a \cdot x\right)\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 65.4%

         \[\leadsto \sinh \left(x \cdot a\right) + \left(-1 \cdot \left(x \cdot \left(-0.5 \cdot a - 0.5 \cdot \color{blue}{a}\right)\right)\right) \cdot \sinh \left(0.5 \cdot \left(a \cdot x\right)\right) \]

   8. Applied rewrites 65.4%

      \[\leadsto \sinh \left(x \cdot a\right) + \color{blue}{\left(-1 \cdot \left(x \cdot \left(-0.5 \cdot a - 0.5 \cdot a\right)\right)\right)} \cdot \sinh \left(0.5 \cdot \left(a \cdot x\right)\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto \sinh \left(x \cdot a\right) + \left(-1 \cdot \left(x \cdot \left(-0.5 \cdot a - 0.5 \cdot a\right)\right)\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(\frac{1}{2} \cdot a - \frac{-1}{2} \cdot a\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 65.8%

          \[\leadsto \sinh \left(x \cdot a\right) + \left(-1 \cdot \left(x \cdot \left(-0.5 \cdot a - 0.5 \cdot a\right)\right)\right) \cdot \left(0.5 \cdot \left(x \cdot \left(0.5 \cdot a - -0.5 \cdot \color{blue}{a}\right)\right)\right) \]

   11. Applied rewrites 65.8%

       \[\leadsto \sinh \left(x \cdot a\right) + \left(-1 \cdot \left(x \cdot \left(-0.5 \cdot a - 0.5 \cdot a\right)\right)\right) \cdot \color{blue}{\left(0.5 \cdot \left(x \cdot \left(0.5 \cdot a - -0.5 \cdot a\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(a, x\right) \cdot \mathsf{max}\left(a, x\right)\\ t_1 := e^{t\_0} - 1\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{max}\left(a, x\right) \cdot \left(\mathsf{min}\left(a, x\right) + 0.5 \cdot \left(t\_0 \cdot \mathsf{min}\left(a, x\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a x)
  :precision binary64
  (let* ((t_0 (* (fmin a x) (fmax a x))) (t_1 (- (exp t_0) 1.0)))
  (if (<= t_1 -4e-5)
    t_1
    (* (fmax a x) (+ (fmin a x) (* 0.5 (* t_0 (fmin a x))))))))

C

double code(double a, double x) {
	double t_0 = fmin(a, x) * fmax(a, x);
	double t_1 = exp(t_0) - 1.0;
	double tmp;
	if (t_1 <= -4e-5) {
		tmp = t_1;
	} else {
		tmp = fmax(a, x) * (fmin(a, x) + (0.5 * (t_0 * fmin(a, 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(a, x)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = fmin(a, x) * fmax(a, x)
    t_1 = exp(t_0) - 1.0d0
    if (t_1 <= (-4d-5)) then
        tmp = t_1
    else
        tmp = fmax(a, x) * (fmin(a, x) + (0.5d0 * (t_0 * fmin(a, x))))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double t_0 = fmin(a, x) * fmax(a, x);
	double t_1 = Math.exp(t_0) - 1.0;
	double tmp;
	if (t_1 <= -4e-5) {
		tmp = t_1;
	} else {
		tmp = fmax(a, x) * (fmin(a, x) + (0.5 * (t_0 * fmin(a, x))));
	}
	return tmp;
}

Python

def code(a, x):
	t_0 = fmin(a, x) * fmax(a, x)
	t_1 = math.exp(t_0) - 1.0
	tmp = 0
	if t_1 <= -4e-5:
		tmp = t_1
	else:
		tmp = fmax(a, x) * (fmin(a, x) + (0.5 * (t_0 * fmin(a, x))))
	return tmp

Julia

function code(a, x)
	t_0 = Float64(fmin(a, x) * fmax(a, x))
	t_1 = Float64(exp(t_0) - 1.0)
	tmp = 0.0
	if (t_1 <= -4e-5)
		tmp = t_1;
	else
		tmp = Float64(fmax(a, x) * Float64(fmin(a, x) + Float64(0.5 * Float64(t_0 * fmin(a, x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	t_0 = min(a, x) * max(a, x);
	t_1 = exp(t_0) - 1.0;
	tmp = 0.0;
	if (t_1 <= -4e-5)
		tmp = t_1;
	else
		tmp = max(a, x) * (min(a, x) + (0.5 * (t_0 * min(a, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := Block[{t$95$0 = N[(N[Min[a, x], $MachinePrecision] * N[Max[a, x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[t$95$0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-5], t$95$1, N[(N[Max[a, x], $MachinePrecision] * N[(N[Min[a, x], $MachinePrecision] + N[(0.5 * N[(t$95$0 * N[Min[a, x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (*.f64 a x)) #s(literal 1 binary64)) < -4.0000000000000003e-5

   1. Initial program 54.8%

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

   if -4.0000000000000003e-5 < (-.f64 (exp.f64 (*.f64 a x)) #s(literal 1 binary64))

   1. Initial program 54.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 61.5%

         \[\leadsto x \cdot \left(a + 0.5 \cdot \left({a}^{2} \cdot x\right)\right) \]

   4. Applied rewrites 61.5%

      \[\leadsto \color{blue}{x \cdot \left(a + 0.5 \cdot \left({a}^{2} \cdot x\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 65.3%

         \[\leadsto x \cdot \left(a + 0.5 \cdot \left(\left(x \cdot a\right) \cdot \color{blue}{a}\right)\right) \]

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 65.3%

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

   6. Applied rewrites 65.3%

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

Alternative 6: 98.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(a, x\right) \cdot \mathsf{max}\left(a, x\right)\\ t_1 := \frac{1}{1 + -0.5 \cdot t\_0}\\ \mathbf{if}\;e^{t\_0} - 1 \leq -1:\\ \;\;\;\;t\_1 \cdot t\_1 - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{max}\left(a, x\right) \cdot \left(\mathsf{min}\left(a, x\right) + 0.5 \cdot \left(t\_0 \cdot \mathsf{min}\left(a, x\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a x)
  :precision binary64
  (let* ((t_0 (* (fmin a x) (fmax a x)))
       (t_1 (/ 1.0 (+ 1.0 (* -0.5 t_0)))))
  (if (<= (- (exp t_0) 1.0) -1.0)
    (- (* t_1 t_1) 1.0)
    (* (fmax a x) (+ (fmin a x) (* 0.5 (* t_0 (fmin a x))))))))

C

double code(double a, double x) {
	double t_0 = fmin(a, x) * fmax(a, x);
	double t_1 = 1.0 / (1.0 + (-0.5 * t_0));
	double tmp;
	if ((exp(t_0) - 1.0) <= -1.0) {
		tmp = (t_1 * t_1) - 1.0;
	} else {
		tmp = fmax(a, x) * (fmin(a, x) + (0.5 * (t_0 * fmin(a, 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(a, x)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = fmin(a, x) * fmax(a, x)
    t_1 = 1.0d0 / (1.0d0 + ((-0.5d0) * t_0))
    if ((exp(t_0) - 1.0d0) <= (-1.0d0)) then
        tmp = (t_1 * t_1) - 1.0d0
    else
        tmp = fmax(a, x) * (fmin(a, x) + (0.5d0 * (t_0 * fmin(a, x))))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double t_0 = fmin(a, x) * fmax(a, x);
	double t_1 = 1.0 / (1.0 + (-0.5 * t_0));
	double tmp;
	if ((Math.exp(t_0) - 1.0) <= -1.0) {
		tmp = (t_1 * t_1) - 1.0;
	} else {
		tmp = fmax(a, x) * (fmin(a, x) + (0.5 * (t_0 * fmin(a, x))));
	}
	return tmp;
}

Python

def code(a, x):
	t_0 = fmin(a, x) * fmax(a, x)
	t_1 = 1.0 / (1.0 + (-0.5 * t_0))
	tmp = 0
	if (math.exp(t_0) - 1.0) <= -1.0:
		tmp = (t_1 * t_1) - 1.0
	else:
		tmp = fmax(a, x) * (fmin(a, x) + (0.5 * (t_0 * fmin(a, x))))
	return tmp

Julia

function code(a, x)
	t_0 = Float64(fmin(a, x) * fmax(a, x))
	t_1 = Float64(1.0 / Float64(1.0 + Float64(-0.5 * t_0)))
	tmp = 0.0
	if (Float64(exp(t_0) - 1.0) <= -1.0)
		tmp = Float64(Float64(t_1 * t_1) - 1.0);
	else
		tmp = Float64(fmax(a, x) * Float64(fmin(a, x) + Float64(0.5 * Float64(t_0 * fmin(a, x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	t_0 = min(a, x) * max(a, x);
	t_1 = 1.0 / (1.0 + (-0.5 * t_0));
	tmp = 0.0;
	if ((exp(t_0) - 1.0) <= -1.0)
		tmp = (t_1 * t_1) - 1.0;
	else
		tmp = max(a, x) * (min(a, x) + (0.5 * (t_0 * min(a, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := Block[{t$95$0 = N[(N[Min[a, x], $MachinePrecision] * N[Max[a, x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(1.0 + N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Exp[t$95$0], $MachinePrecision] - 1.0), $MachinePrecision], -1.0], N[(N[(t$95$1 * t$95$1), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[Max[a, x], $MachinePrecision] * N[(N[Min[a, x], $MachinePrecision] + N[(0.5 * N[(t$95$0 * N[Min[a, x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 54.8%

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

      1. lift-exp.f64 N/A

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

      2. exp-fabs N/A

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

      3. lift-exp.f64 N/A

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

      4. rem-sqrt-square-rev N/A

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

      5. sqrt-prod N/A

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

      6. lower-unsound-*.f64 N/A

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

      7. lower-unsound-sqrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-unsound-sqrt.f64 54.8%

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 54.8%

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

   3. Applied rewrites 54.8%

      \[\leadsto \color{blue}{\sqrt{e^{x \cdot a}} \cdot \sqrt{e^{x \cdot a}}} - 1 \]
   4. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. exp-sqrt-rev N/A

         \[\leadsto \color{blue}{e^{\frac{x \cdot a}{2}}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      4. +-lft-identity N/A

         \[\leadsto e^{\frac{\color{blue}{0 + x \cdot a}}{2}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      5. lift-*.f64 N/A

         \[\leadsto e^{\frac{0 + \color{blue}{x \cdot a}}{2}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

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

         \[\leadsto e^{\frac{\color{blue}{0 - \left(\mathsf{neg}\left(x\right)\right) \cdot a}}{2}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      7. lift-neg.f64 N/A

         \[\leadsto e^{\frac{0 - \color{blue}{\left(-x\right)} \cdot a}{2}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      8. lift-*.f64 N/A

         \[\leadsto e^{\frac{0 - \color{blue}{\left(-x\right) \cdot a}}{2}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      9. div-sub N/A

         \[\leadsto e^{\color{blue}{\frac{0}{2} - \frac{\left(-x\right) \cdot a}{2}}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      10. metadata-eval N/A

          \[\leadsto e^{\color{blue}{0} - \frac{\left(-x\right) \cdot a}{2}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      11. mult-flip-rev N/A

          \[\leadsto e^{0 - \color{blue}{\left(\left(-x\right) \cdot a\right) \cdot \frac{1}{2}}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      12. lift-*.f64 N/A

          \[\leadsto e^{0 - \color{blue}{\left(\left(-x\right) \cdot a\right)} \cdot \frac{1}{2}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      13. lift-neg.f64 N/A

          \[\leadsto e^{0 - \left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot a\right) \cdot \frac{1}{2}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      14. distribute-lft-neg-out N/A

          \[\leadsto e^{0 - \color{blue}{\left(\mathsf{neg}\left(x \cdot a\right)\right)} \cdot \frac{1}{2}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      15. lift-*.f64 N/A

          \[\leadsto e^{0 - \left(\mathsf{neg}\left(\color{blue}{x \cdot a}\right)\right) \cdot \frac{1}{2}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      16. sub0-neg N/A

          \[\leadsto e^{0 - \color{blue}{\left(0 - x \cdot a\right)} \cdot \frac{1}{2}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      17. lift--.f64 N/A

          \[\leadsto e^{0 - \color{blue}{\left(0 - x \cdot a\right)} \cdot \frac{1}{2}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      18. metadata-eval N/A

          \[\leadsto e^{0 - \left(0 - x \cdot a\right) \cdot \color{blue}{\frac{1}{2}}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      19. lift-*.f64 N/A

          \[\leadsto e^{0 - \color{blue}{\left(0 - x \cdot a\right) \cdot \frac{1}{2}}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      20. div-exp N/A

          \[\leadsto \color{blue}{\frac{e^{0}}{e^{\left(0 - x \cdot a\right) \cdot \frac{1}{2}}}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      21. 1-exp N/A

          \[\leadsto \frac{\color{blue}{1}}{e^{\left(0 - x \cdot a\right) \cdot \frac{1}{2}}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

      22. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{e^{\left(0 - x \cdot a\right) \cdot \frac{1}{2}}}} \cdot \sqrt{e^{x \cdot a}} - 1 \]

   5. Applied rewrites 54.8%

      \[\leadsto \color{blue}{\frac{1}{e^{-0.5 \cdot \left(a \cdot x\right)}}} \cdot \sqrt{e^{x \cdot a}} - 1 \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. exp-sqrt-rev N/A

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

      4. +-lft-identity N/A

         \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{\frac{\color{blue}{0 + x \cdot a}}{2}} - 1 \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{\frac{0 + \color{blue}{x \cdot a}}{2}} - 1 \]

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

         \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{\frac{\color{blue}{0 - \left(\mathsf{neg}\left(x\right)\right) \cdot a}}{2}} - 1 \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{\frac{0 - \color{blue}{\left(-x\right)} \cdot a}{2}} - 1 \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{\frac{0 - \color{blue}{\left(-x\right) \cdot a}}{2}} - 1 \]

      9. div-sub N/A

         \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{\color{blue}{\frac{0}{2} - \frac{\left(-x\right) \cdot a}{2}}} - 1 \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{\color{blue}{0} - \frac{\left(-x\right) \cdot a}{2}} - 1 \]

      11. mult-flip-rev N/A

          \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{0 - \color{blue}{\left(\left(-x\right) \cdot a\right) \cdot \frac{1}{2}}} - 1 \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{0 - \color{blue}{\left(\left(-x\right) \cdot a\right)} \cdot \frac{1}{2}} - 1 \]

      13. lift-neg.f64 N/A

          \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{0 - \left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot a\right) \cdot \frac{1}{2}} - 1 \]

      14. distribute-lft-neg-out N/A

          \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{0 - \color{blue}{\left(\mathsf{neg}\left(x \cdot a\right)\right)} \cdot \frac{1}{2}} - 1 \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{0 - \left(\mathsf{neg}\left(\color{blue}{x \cdot a}\right)\right) \cdot \frac{1}{2}} - 1 \]

      16. sub0-neg N/A

          \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{0 - \color{blue}{\left(0 - x \cdot a\right)} \cdot \frac{1}{2}} - 1 \]

      17. lift--.f64 N/A

          \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{0 - \color{blue}{\left(0 - x \cdot a\right)} \cdot \frac{1}{2}} - 1 \]

      18. metadata-eval N/A

          \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{0 - \left(0 - x \cdot a\right) \cdot \color{blue}{\frac{1}{2}}} - 1 \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot e^{0 - \color{blue}{\left(0 - x \cdot a\right) \cdot \frac{1}{2}}} - 1 \]

      20. div-exp N/A

          \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot \color{blue}{\frac{e^{0}}{e^{\left(0 - x \cdot a\right) \cdot \frac{1}{2}}}} - 1 \]

      21. 1-exp N/A

          \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot \frac{\color{blue}{1}}{e^{\left(0 - x \cdot a\right) \cdot \frac{1}{2}}} - 1 \]

      22. lower-/.f64 N/A

          \[\leadsto \frac{1}{e^{\frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot \color{blue}{\frac{1}{e^{\left(0 - x \cdot a\right) \cdot \frac{1}{2}}}} - 1 \]

   7. Applied rewrites 54.8%

      \[\leadsto \frac{1}{e^{-0.5 \cdot \left(a \cdot x\right)}} \cdot \color{blue}{\frac{1}{e^{-0.5 \cdot \left(a \cdot x\right)}}} - 1 \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{1}{\color{blue}{1 + \frac{-1}{2} \cdot \left(a \cdot x\right)}} \cdot \frac{1}{e^{-0.5 \cdot \left(a \cdot x\right)}} - 1 \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 53.6%

         \[\leadsto \frac{1}{1 + -0.5 \cdot \left(a \cdot \color{blue}{x}\right)} \cdot \frac{1}{e^{-0.5 \cdot \left(a \cdot x\right)}} - 1 \]

   10. Applied rewrites 53.6%

       \[\leadsto \frac{1}{\color{blue}{1 + -0.5 \cdot \left(a \cdot x\right)}} \cdot \frac{1}{e^{-0.5 \cdot \left(a \cdot x\right)}} - 1 \]

   11. Taylor expanded in a around 0

       \[\leadsto \frac{1}{1 + -0.5 \cdot \left(a \cdot x\right)} \cdot \frac{1}{\color{blue}{1 + \frac{-1}{2} \cdot \left(a \cdot x\right)}} - 1 \]
   12. Step-by-step derivation

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 53.3%

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

   13. Applied rewrites 53.3%

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

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

   1. Initial program 54.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 61.5%

         \[\leadsto x \cdot \left(a + 0.5 \cdot \left({a}^{2} \cdot x\right)\right) \]

   4. Applied rewrites 61.5%

      \[\leadsto \color{blue}{x \cdot \left(a + 0.5 \cdot \left({a}^{2} \cdot x\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 65.3%

         \[\leadsto x \cdot \left(a + 0.5 \cdot \left(\left(x \cdot a\right) \cdot \color{blue}{a}\right)\right) \]

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 65.3%

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

   6. Applied rewrites 65.3%

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

Alternative 7: 68.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(a, x\right) \cdot \mathsf{max}\left(a, x\right)\\ \mathbf{if}\;e^{t\_0} - 1 \leq -1:\\ \;\;\;\;\mathsf{min}\left(a, x\right) \cdot \frac{\left(\left(t\_0 - -1\right) \cdot \mathsf{min}\left(a, x\right)\right) \cdot 1}{\mathsf{min}\left(a, x\right) \cdot \mathsf{min}\left(a, x\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{max}\left(a, x\right) \cdot \left(\mathsf{min}\left(a, x\right) + 0.5 \cdot \left(t\_0 \cdot \mathsf{min}\left(a, x\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a x)
  :precision binary64
  (let* ((t_0 (* (fmin a x) (fmax a x))))
  (if (<= (- (exp t_0) 1.0) -1.0)
    (-
     (*
      (fmin a x)
      (/
       (* (* (- t_0 -1.0) (fmin a x)) 1.0)
       (* (fmin a x) (fmin a x))))
     1.0)
    (* (fmax a x) (+ (fmin a x) (* 0.5 (* t_0 (fmin a x))))))))

C

double code(double a, double x) {
	double t_0 = fmin(a, x) * fmax(a, x);
	double tmp;
	if ((exp(t_0) - 1.0) <= -1.0) {
		tmp = (fmin(a, x) * ((((t_0 - -1.0) * fmin(a, x)) * 1.0) / (fmin(a, x) * fmin(a, x)))) - 1.0;
	} else {
		tmp = fmax(a, x) * (fmin(a, x) + (0.5 * (t_0 * fmin(a, 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(a, x)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = fmin(a, x) * fmax(a, x)
    if ((exp(t_0) - 1.0d0) <= (-1.0d0)) then
        tmp = (fmin(a, x) * ((((t_0 - (-1.0d0)) * fmin(a, x)) * 1.0d0) / (fmin(a, x) * fmin(a, x)))) - 1.0d0
    else
        tmp = fmax(a, x) * (fmin(a, x) + (0.5d0 * (t_0 * fmin(a, x))))
    end if
    code = tmp
end function

Java

public static double code(double a, double x) {
	double t_0 = fmin(a, x) * fmax(a, x);
	double tmp;
	if ((Math.exp(t_0) - 1.0) <= -1.0) {
		tmp = (fmin(a, x) * ((((t_0 - -1.0) * fmin(a, x)) * 1.0) / (fmin(a, x) * fmin(a, x)))) - 1.0;
	} else {
		tmp = fmax(a, x) * (fmin(a, x) + (0.5 * (t_0 * fmin(a, x))));
	}
	return tmp;
}

Python

def code(a, x):
	t_0 = fmin(a, x) * fmax(a, x)
	tmp = 0
	if (math.exp(t_0) - 1.0) <= -1.0:
		tmp = (fmin(a, x) * ((((t_0 - -1.0) * fmin(a, x)) * 1.0) / (fmin(a, x) * fmin(a, x)))) - 1.0
	else:
		tmp = fmax(a, x) * (fmin(a, x) + (0.5 * (t_0 * fmin(a, x))))
	return tmp

Julia

function code(a, x)
	t_0 = Float64(fmin(a, x) * fmax(a, x))
	tmp = 0.0
	if (Float64(exp(t_0) - 1.0) <= -1.0)
		tmp = Float64(Float64(fmin(a, x) * Float64(Float64(Float64(Float64(t_0 - -1.0) * fmin(a, x)) * 1.0) / Float64(fmin(a, x) * fmin(a, x)))) - 1.0);
	else
		tmp = Float64(fmax(a, x) * Float64(fmin(a, x) + Float64(0.5 * Float64(t_0 * fmin(a, x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, x)
	t_0 = min(a, x) * max(a, x);
	tmp = 0.0;
	if ((exp(t_0) - 1.0) <= -1.0)
		tmp = (min(a, x) * ((((t_0 - -1.0) * min(a, x)) * 1.0) / (min(a, x) * min(a, x)))) - 1.0;
	else
		tmp = max(a, x) * (min(a, x) + (0.5 * (t_0 * min(a, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, x_] := Block[{t$95$0 = N[(N[Min[a, x], $MachinePrecision] * N[Max[a, x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Exp[t$95$0], $MachinePrecision] - 1.0), $MachinePrecision], -1.0], N[(N[(N[Min[a, x], $MachinePrecision] * N[(N[(N[(N[(t$95$0 - -1.0), $MachinePrecision] * N[Min[a, x], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] / N[(N[Min[a, x], $MachinePrecision] * N[Min[a, x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[Max[a, x], $MachinePrecision] * N[(N[Min[a, x], $MachinePrecision] + N[(0.5 * N[(t$95$0 * N[Min[a, x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 54.8%

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

   2. Taylor expanded in a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 21.5%

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

   4. Applied rewrites 21.5%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 18.9%

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

   7. Applied rewrites 18.9%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

         \[\leadsto a \cdot \frac{x \cdot a + 1}{a} - 1 \]

      4. *-commutative N/A

         \[\leadsto a \cdot \frac{a \cdot x + 1}{a} - 1 \]

      5. +-commutative N/A

         \[\leadsto a \cdot \frac{1 + a \cdot x}{a} - 1 \]

      6. mult-flip N/A

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

      7. +-commutative N/A

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

      8. rgt-mult-inverse N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-lft-in N/A

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

      11. lift-+.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. add-to-fraction N/A

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

      16. *-commutative N/A

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

      17. +-commutative N/A

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

      18. associate-*l/ N/A

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

      19. frac-times N/A

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

      20. lift-*.f64 N/A

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

      21. lower-/.f64 N/A

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

   9. Applied rewrites 8.7%

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

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

   1. Initial program 54.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 61.5%

         \[\leadsto x \cdot \left(a + 0.5 \cdot \left({a}^{2} \cdot x\right)\right) \]

   4. Applied rewrites 61.5%

      \[\leadsto \color{blue}{x \cdot \left(a + 0.5 \cdot \left({a}^{2} \cdot x\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 65.3%

         \[\leadsto x \cdot \left(a + 0.5 \cdot \left(\left(x \cdot a\right) \cdot \color{blue}{a}\right)\right) \]

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 65.3%

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

   6. Applied rewrites 65.3%

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

Alternative 8: 65.3% accurate, 0.2× speedup

Math

\[\mathsf{max}\left(a, x\right) \cdot \left(\mathsf{min}\left(a, x\right) + 0.5 \cdot \left(\left(\mathsf{min}\left(a, x\right) \cdot \mathsf{max}\left(a, x\right)\right) \cdot \mathsf{min}\left(a, x\right)\right)\right) \]

FPCore

(FPCore (a x)
  :precision binary64
  (*
 (fmax a x)
 (+ (fmin a x) (* 0.5 (* (* (fmin a x) (fmax a x)) (fmin a x))))))

C

double code(double a, double x) {
	return fmax(a, x) * (fmin(a, x) + (0.5 * ((fmin(a, x) * fmax(a, x)) * fmin(a, 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(a, x)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: x
    code = fmax(a, x) * (fmin(a, x) + (0.5d0 * ((fmin(a, x) * fmax(a, x)) * fmin(a, x))))
end function

Java

public static double code(double a, double x) {
	return fmax(a, x) * (fmin(a, x) + (0.5 * ((fmin(a, x) * fmax(a, x)) * fmin(a, x))));
}

Python

def code(a, x):
	return fmax(a, x) * (fmin(a, x) + (0.5 * ((fmin(a, x) * fmax(a, x)) * fmin(a, x))))

Julia

function code(a, x)
	return Float64(fmax(a, x) * Float64(fmin(a, x) + Float64(0.5 * Float64(Float64(fmin(a, x) * fmax(a, x)) * fmin(a, x)))))
end

MATLAB

function tmp = code(a, x)
	tmp = max(a, x) * (min(a, x) + (0.5 * ((min(a, x) * max(a, x)) * min(a, x))));
end

Wolfram

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

Derivation

1. Initial program 54.8%

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

2. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 61.5%

      \[\leadsto x \cdot \left(a + 0.5 \cdot \left({a}^{2} \cdot x\right)\right) \]

4. Applied rewrites 61.5%

   \[\leadsto \color{blue}{x \cdot \left(a + 0.5 \cdot \left({a}^{2} \cdot x\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f64 N/A

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

   7. lower-*.f64 65.3%

      \[\leadsto x \cdot \left(a + 0.5 \cdot \left(\left(x \cdot a\right) \cdot \color{blue}{a}\right)\right) \]

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 65.3%

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

6. Applied rewrites 65.3%

   \[\leadsto x \cdot \left(a + 0.5 \cdot \left(\left(a \cdot x\right) \cdot \color{blue}{a}\right)\right) \]
7. Add Preprocessing

Alternative 9: 21.5% accurate, 9.1× speedup

Math

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

FPCore

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

C

double code(double a, double x) {
	return (1.0 + (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 = (1.0d0 + (a * x)) - 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

2. Taylor expanded in a around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 21.5%

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

4. Applied rewrites 21.5%

   \[\leadsto \color{blue}{\left(1 + a \cdot x\right)} - 1 \]
5. Add Preprocessing

Alternative 10: 19.7% accurate, 27.3× speedup

Math

\[1 - 1 \]

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 54.8%

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

2. Taylor expanded in a around 0

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

4. Applied rewrites 19.7%

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

Reproduce

herbie shell --seed 2025258 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :pre (> 710.0 (* a x))
  (- (exp (* a x)) 1.0))