expq3 (problem 3.4.2)

Percentage Accurate: 0.0% → 79.7%

Time: 14.9s

Alternatives: 3

Speedup: 17.5×

Specification

\[\left(\left|a\right| \leq 710 \land \left|b\right| \leq 710\right) \land \left(10^{-27} \cdot \mathsf{min}\left(\left|a\right|, \left|b\right|\right) \leq \varepsilon \land \varepsilon \leq \mathsf{min}\left(\left|a\right|, \left|b\right|\right)\right)\]

Math

\[\begin{array}{l} \\ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \end{array} \]

FPCore

(FPCore (a b eps)
 :precision binary64
 (/
  (* eps (- (exp (* (+ a b) eps)) 1.0))
  (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))

C

double code(double a, double b, double eps) {
	return (eps * (exp(((a + b) * eps)) - 1.0)) / ((exp((a * eps)) - 1.0) * (exp((b * eps)) - 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, b, eps)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: eps
    code = (eps * (exp(((a + b) * eps)) - 1.0d0)) / ((exp((a * eps)) - 1.0d0) * (exp((b * eps)) - 1.0d0))
end function

Java

public static double code(double a, double b, double eps) {
	return (eps * (Math.exp(((a + b) * eps)) - 1.0)) / ((Math.exp((a * eps)) - 1.0) * (Math.exp((b * eps)) - 1.0));
}

Python

def code(a, b, eps):
	return (eps * (math.exp(((a + b) * eps)) - 1.0)) / ((math.exp((a * eps)) - 1.0) * (math.exp((b * eps)) - 1.0))

Julia

function code(a, b, eps)
	return Float64(Float64(eps * Float64(exp(Float64(Float64(a + b) * eps)) - 1.0)) / Float64(Float64(exp(Float64(a * eps)) - 1.0) * Float64(exp(Float64(b * eps)) - 1.0)))
end

MATLAB

function tmp = code(a, b, eps)
	tmp = (eps * (exp(((a + b) * eps)) - 1.0)) / ((exp((a * eps)) - 1.0) * (exp((b * eps)) - 1.0));
end

Wolfram

code[a_, b_, eps_] := N[(N[(eps * N[(N[Exp[N[(N[(a + b), $MachinePrecision] * eps), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Exp[N[(a * eps), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[Exp[N[(b * eps), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 0.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \end{array} \]

FPCore

(FPCore (a b eps)
 :precision binary64
 (/
  (* eps (- (exp (* (+ a b) eps)) 1.0))
  (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))

C

double code(double a, double b, double eps) {
	return (eps * (exp(((a + b) * eps)) - 1.0)) / ((exp((a * eps)) - 1.0) * (exp((b * eps)) - 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, b, eps)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: eps
    code = (eps * (exp(((a + b) * eps)) - 1.0d0)) / ((exp((a * eps)) - 1.0d0) * (exp((b * eps)) - 1.0d0))
end function

Java

public static double code(double a, double b, double eps) {
	return (eps * (Math.exp(((a + b) * eps)) - 1.0)) / ((Math.exp((a * eps)) - 1.0) * (Math.exp((b * eps)) - 1.0));
}

Python

def code(a, b, eps):
	return (eps * (math.exp(((a + b) * eps)) - 1.0)) / ((math.exp((a * eps)) - 1.0) * (math.exp((b * eps)) - 1.0))

Julia

function code(a, b, eps)
	return Float64(Float64(eps * Float64(exp(Float64(Float64(a + b) * eps)) - 1.0)) / Float64(Float64(exp(Float64(a * eps)) - 1.0) * Float64(exp(Float64(b * eps)) - 1.0)))
end

MATLAB

function tmp = code(a, b, eps)
	tmp = (eps * (exp(((a + b) * eps)) - 1.0)) / ((exp((a * eps)) - 1.0) * (exp((b * eps)) - 1.0));
end

Wolfram

code[a_, b_, eps_] := N[(N[(eps * N[(N[Exp[N[(N[(a + b), $MachinePrecision] * eps), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Exp[N[(a * eps), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[Exp[N[(b * eps), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 79.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} [a, b, eps] = \mathsf{sort}([a, b, eps])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 10^{-150}:\\ \;\;\;\;{b}^{-1}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-44}:\\ \;\;\;\;{a}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b}{a \cdot b}\\ \end{array} \end{array} \]

FPCore

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
(FPCore (a b eps)
 :precision binary64
 (if (<= b 1e-150)
   (pow b -1.0)
   (if (<= b 2.9e-44) (pow a -1.0) (/ (+ a b) (* a b)))))

C

assert(a < b && b < eps);
double code(double a, double b, double eps) {
	double tmp;
	if (b <= 1e-150) {
		tmp = pow(b, -1.0);
	} else if (b <= 2.9e-44) {
		tmp = pow(a, -1.0);
	} else {
		tmp = (a + b) / (a * b);
	}
	return tmp;
}

Fortran

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
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, b, eps)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (b <= 1d-150) then
        tmp = b ** (-1.0d0)
    else if (b <= 2.9d-44) then
        tmp = a ** (-1.0d0)
    else
        tmp = (a + b) / (a * b)
    end if
    code = tmp
end function

Java

assert a < b && b < eps;
public static double code(double a, double b, double eps) {
	double tmp;
	if (b <= 1e-150) {
		tmp = Math.pow(b, -1.0);
	} else if (b <= 2.9e-44) {
		tmp = Math.pow(a, -1.0);
	} else {
		tmp = (a + b) / (a * b);
	}
	return tmp;
}

Python

[a, b, eps] = sort([a, b, eps])
def code(a, b, eps):
	tmp = 0
	if b <= 1e-150:
		tmp = math.pow(b, -1.0)
	elif b <= 2.9e-44:
		tmp = math.pow(a, -1.0)
	else:
		tmp = (a + b) / (a * b)
	return tmp

Julia

a, b, eps = sort([a, b, eps])
function code(a, b, eps)
	tmp = 0.0
	if (b <= 1e-150)
		tmp = b ^ -1.0;
	elseif (b <= 2.9e-44)
		tmp = a ^ -1.0;
	else
		tmp = Float64(Float64(a + b) / Float64(a * b));
	end
	return tmp
end

MATLAB

a, b, eps = num2cell(sort([a, b, eps])){:}
function tmp_2 = code(a, b, eps)
	tmp = 0.0;
	if (b <= 1e-150)
		tmp = b ^ -1.0;
	elseif (b <= 2.9e-44)
		tmp = a ^ -1.0;
	else
		tmp = (a + b) / (a * b);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
code[a_, b_, eps_] := If[LessEqual[b, 1e-150], N[Power[b, -1.0], $MachinePrecision], If[LessEqual[b, 2.9e-44], N[Power[a, -1.0], $MachinePrecision], N[(N[(a + b), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.00000000000000001e-150

   1. Initial program 0.0%

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 0.0

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lower-expm1.f64 0.3

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)} \cdot \varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\left(a + b\right)} \cdot \varepsilon\right) \cdot \varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\left(b + a\right)} \cdot \varepsilon\right) \cdot \varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      9. lower-+.f64 0.3

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\left(b + a\right)} \cdot \varepsilon\right) \cdot \varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\left(e^{b \cdot \varepsilon} - 1\right) \cdot \left(e^{a \cdot \varepsilon} - 1\right)}} \]

      12. lower-*.f64 0.3

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\left(e^{b \cdot \varepsilon} - 1\right) \cdot \left(e^{a \cdot \varepsilon} - 1\right)}} \]

      13. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\left(e^{b \cdot \varepsilon} - 1\right)} \cdot \left(e^{a \cdot \varepsilon} - 1\right)} \]

      14. lift-exp.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\left(\color{blue}{e^{b \cdot \varepsilon}} - 1\right) \cdot \left(e^{a \cdot \varepsilon} - 1\right)} \]

      15. lower-expm1.f64 0.1

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)} \cdot \left(e^{a \cdot \varepsilon} - 1\right)} \]

      16. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \color{blue}{\left(e^{a \cdot \varepsilon} - 1\right)}} \]

      17. lift-exp.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \left(\color{blue}{e^{a \cdot \varepsilon}} - 1\right)} \]

      18. lower-expm1.f64 4.3

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}} \]

   3. Applied rewrites 4.3%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\left(b + a\right) \cdot \varepsilon}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(b + a\right)}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\varepsilon \cdot \color{blue}{\left(b + a\right)}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b + \varepsilon \cdot a}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{b \cdot \varepsilon} + \varepsilon \cdot a\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{b \cdot \varepsilon} + \varepsilon \cdot a\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a + b \cdot \varepsilon}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      8. lift-fma.f64 4.3

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\varepsilon, a, b \cdot \varepsilon\right)}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\varepsilon, a, \color{blue}{b \cdot \varepsilon}\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\varepsilon, a, \color{blue}{\varepsilon \cdot b}\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      11. lower-*.f64 4.3

          \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\varepsilon, a, \color{blue}{\varepsilon \cdot b}\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

   5. Applied rewrites 4.3%

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\varepsilon, a, \varepsilon \cdot b\right)}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{1}{b}} \]
   7. Step-by-step derivation

      1. inv-pow N/A

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

      2. lower-pow.f64 76.5

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

   8. Applied rewrites 76.5%

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

   if 1.00000000000000001e-150 < b < 2.9000000000000001e-44

   1. Initial program 0.0%

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 0.0

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lower-expm1.f64 0.2

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)} \cdot \varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\left(a + b\right)} \cdot \varepsilon\right) \cdot \varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\left(b + a\right)} \cdot \varepsilon\right) \cdot \varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      9. lower-+.f64 0.2

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\left(b + a\right)} \cdot \varepsilon\right) \cdot \varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\left(e^{b \cdot \varepsilon} - 1\right) \cdot \left(e^{a \cdot \varepsilon} - 1\right)}} \]

      12. lower-*.f64 0.2

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\left(e^{b \cdot \varepsilon} - 1\right) \cdot \left(e^{a \cdot \varepsilon} - 1\right)}} \]

      13. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\left(e^{b \cdot \varepsilon} - 1\right)} \cdot \left(e^{a \cdot \varepsilon} - 1\right)} \]

      14. lift-exp.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\left(\color{blue}{e^{b \cdot \varepsilon}} - 1\right) \cdot \left(e^{a \cdot \varepsilon} - 1\right)} \]

      15. lower-expm1.f64 0.2

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)} \cdot \left(e^{a \cdot \varepsilon} - 1\right)} \]

      16. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \color{blue}{\left(e^{a \cdot \varepsilon} - 1\right)}} \]

      17. lift-exp.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \left(\color{blue}{e^{a \cdot \varepsilon}} - 1\right)} \]

      18. lower-expm1.f64 7.7

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}} \]

   3. Applied rewrites 7.7%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\left(b + a\right) \cdot \varepsilon}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(b + a\right)}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\varepsilon \cdot \color{blue}{\left(b + a\right)}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b + \varepsilon \cdot a}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{b \cdot \varepsilon} + \varepsilon \cdot a\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{b \cdot \varepsilon} + \varepsilon \cdot a\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a + b \cdot \varepsilon}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      8. lift-fma.f64 7.7

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\varepsilon, a, b \cdot \varepsilon\right)}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\varepsilon, a, \color{blue}{b \cdot \varepsilon}\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\varepsilon, a, \color{blue}{\varepsilon \cdot b}\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      11. lower-*.f64 7.7

          \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\varepsilon, a, \color{blue}{\varepsilon \cdot b}\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

   5. Applied rewrites 7.7%

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\varepsilon, a, \varepsilon \cdot b\right)}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{a}} \]
   7. Step-by-step derivation

      1. inv-pow N/A

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

      2. lower-pow.f64 76.9

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

   8. Applied rewrites 76.9%

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

   if 2.9000000000000001e-44 < b

   1. Initial program 0.2%

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 0.2

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lower-expm1.f64 1.1

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)} \cdot \varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\left(a + b\right)} \cdot \varepsilon\right) \cdot \varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\left(b + a\right)} \cdot \varepsilon\right) \cdot \varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      9. lower-+.f64 1.1

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\left(b + a\right)} \cdot \varepsilon\right) \cdot \varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\left(e^{b \cdot \varepsilon} - 1\right) \cdot \left(e^{a \cdot \varepsilon} - 1\right)}} \]

      12. lower-*.f64 1.1

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\left(e^{b \cdot \varepsilon} - 1\right) \cdot \left(e^{a \cdot \varepsilon} - 1\right)}} \]

      13. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\left(e^{b \cdot \varepsilon} - 1\right)} \cdot \left(e^{a \cdot \varepsilon} - 1\right)} \]

      14. lift-exp.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\left(\color{blue}{e^{b \cdot \varepsilon}} - 1\right) \cdot \left(e^{a \cdot \varepsilon} - 1\right)} \]

      15. lower-expm1.f64 1.1

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)} \cdot \left(e^{a \cdot \varepsilon} - 1\right)} \]

      16. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \color{blue}{\left(e^{a \cdot \varepsilon} - 1\right)}} \]

      17. lift-exp.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \left(\color{blue}{e^{a \cdot \varepsilon}} - 1\right)} \]

      18. lower-expm1.f64 29.1

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}} \]

   3. Applied rewrites 29.1%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\left(b + a\right) \cdot \varepsilon}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(b + a\right)}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\varepsilon \cdot \color{blue}{\left(b + a\right)}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b + \varepsilon \cdot a}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{b \cdot \varepsilon} + \varepsilon \cdot a\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{b \cdot \varepsilon} + \varepsilon \cdot a\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a + b \cdot \varepsilon}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      8. lift-fma.f64 29.2

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\varepsilon, a, b \cdot \varepsilon\right)}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\varepsilon, a, \color{blue}{b \cdot \varepsilon}\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\varepsilon, a, \color{blue}{\varepsilon \cdot b}\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      11. lower-*.f64 29.2

          \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\varepsilon, a, \color{blue}{\varepsilon \cdot b}\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

   5. Applied rewrites 29.2%

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\varepsilon, a, \varepsilon \cdot b\right)}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

   6. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\frac{a + b}{a \cdot b}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{a + b}{\color{blue}{a \cdot b}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{a + b}{\color{blue}{a} \cdot b} \]

      3. lower-*.f64 97.7

         \[\leadsto \frac{a + b}{a \cdot \color{blue}{b}} \]

   8. Applied rewrites 97.7%

      \[\leadsto \color{blue}{\frac{a + b}{a \cdot b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 78.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} [a, b, eps] = \mathsf{sort}([a, b, eps])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -6.9 \cdot 10^{-162}:\\ \;\;\;\;\frac{a + b}{a \cdot b}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-1}\\ \end{array} \end{array} \]

FPCore

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
(FPCore (a b eps)
 :precision binary64
 (if (<= a -6.9e-162) (/ (+ a b) (* a b)) (pow a -1.0)))

C

assert(a < b && b < eps);
double code(double a, double b, double eps) {
	double tmp;
	if (a <= -6.9e-162) {
		tmp = (a + b) / (a * b);
	} else {
		tmp = pow(a, -1.0);
	}
	return tmp;
}

Fortran

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
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, b, eps)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (a <= (-6.9d-162)) then
        tmp = (a + b) / (a * b)
    else
        tmp = a ** (-1.0d0)
    end if
    code = tmp
end function

Java

assert a < b && b < eps;
public static double code(double a, double b, double eps) {
	double tmp;
	if (a <= -6.9e-162) {
		tmp = (a + b) / (a * b);
	} else {
		tmp = Math.pow(a, -1.0);
	}
	return tmp;
}

Python

[a, b, eps] = sort([a, b, eps])
def code(a, b, eps):
	tmp = 0
	if a <= -6.9e-162:
		tmp = (a + b) / (a * b)
	else:
		tmp = math.pow(a, -1.0)
	return tmp

Julia

a, b, eps = sort([a, b, eps])
function code(a, b, eps)
	tmp = 0.0
	if (a <= -6.9e-162)
		tmp = Float64(Float64(a + b) / Float64(a * b));
	else
		tmp = a ^ -1.0;
	end
	return tmp
end

MATLAB

a, b, eps = num2cell(sort([a, b, eps])){:}
function tmp_2 = code(a, b, eps)
	tmp = 0.0;
	if (a <= -6.9e-162)
		tmp = (a + b) / (a * b);
	else
		tmp = a ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
code[a_, b_, eps_] := If[LessEqual[a, -6.9e-162], N[(N[(a + b), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision], N[Power[a, -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.9000000000000004e-162

   1. Initial program 0.0%

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 0.0

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lower-expm1.f64 0.5

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)} \cdot \varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\left(a + b\right)} \cdot \varepsilon\right) \cdot \varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\left(b + a\right)} \cdot \varepsilon\right) \cdot \varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      9. lower-+.f64 0.5

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\left(b + a\right)} \cdot \varepsilon\right) \cdot \varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\left(e^{b \cdot \varepsilon} - 1\right) \cdot \left(e^{a \cdot \varepsilon} - 1\right)}} \]

      12. lower-*.f64 0.5

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\left(e^{b \cdot \varepsilon} - 1\right) \cdot \left(e^{a \cdot \varepsilon} - 1\right)}} \]

      13. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\left(e^{b \cdot \varepsilon} - 1\right)} \cdot \left(e^{a \cdot \varepsilon} - 1\right)} \]

      14. lift-exp.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\left(\color{blue}{e^{b \cdot \varepsilon}} - 1\right) \cdot \left(e^{a \cdot \varepsilon} - 1\right)} \]

      15. lower-expm1.f64 0.2

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)} \cdot \left(e^{a \cdot \varepsilon} - 1\right)} \]

      16. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \color{blue}{\left(e^{a \cdot \varepsilon} - 1\right)}} \]

      17. lift-exp.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \left(\color{blue}{e^{a \cdot \varepsilon}} - 1\right)} \]

      18. lower-expm1.f64 14.1

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}} \]

   3. Applied rewrites 14.1%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\left(b + a\right) \cdot \varepsilon}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(b + a\right)}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\varepsilon \cdot \color{blue}{\left(b + a\right)}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b + \varepsilon \cdot a}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{b \cdot \varepsilon} + \varepsilon \cdot a\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{b \cdot \varepsilon} + \varepsilon \cdot a\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a + b \cdot \varepsilon}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      8. lift-fma.f64 14.1

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\varepsilon, a, b \cdot \varepsilon\right)}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\varepsilon, a, \color{blue}{b \cdot \varepsilon}\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\varepsilon, a, \color{blue}{\varepsilon \cdot b}\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      11. lower-*.f64 14.1

          \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\varepsilon, a, \color{blue}{\varepsilon \cdot b}\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

   5. Applied rewrites 14.1%

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\varepsilon, a, \varepsilon \cdot b\right)}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

   6. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\frac{a + b}{a \cdot b}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{a + b}{\color{blue}{a \cdot b}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{a + b}{\color{blue}{a} \cdot b} \]

      3. lower-*.f64 78.9

         \[\leadsto \frac{a + b}{a \cdot \color{blue}{b}} \]

   8. Applied rewrites 78.9%

      \[\leadsto \color{blue}{\frac{a + b}{a \cdot b}} \]

   if -6.9000000000000004e-162 < a

   1. Initial program 0.0%

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 0.0

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

      4. lift--.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. lower-expm1.f64 0.3

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)} \cdot \varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\left(a + b\right)} \cdot \varepsilon\right) \cdot \varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\left(b + a\right)} \cdot \varepsilon\right) \cdot \varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      9. lower-+.f64 0.3

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\left(b + a\right)} \cdot \varepsilon\right) \cdot \varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\left(e^{b \cdot \varepsilon} - 1\right) \cdot \left(e^{a \cdot \varepsilon} - 1\right)}} \]

      12. lower-*.f64 0.3

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\left(e^{b \cdot \varepsilon} - 1\right) \cdot \left(e^{a \cdot \varepsilon} - 1\right)}} \]

      13. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\left(e^{b \cdot \varepsilon} - 1\right)} \cdot \left(e^{a \cdot \varepsilon} - 1\right)} \]

      14. lift-exp.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\left(\color{blue}{e^{b \cdot \varepsilon}} - 1\right) \cdot \left(e^{a \cdot \varepsilon} - 1\right)} \]

      15. lower-expm1.f64 0.3

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)} \cdot \left(e^{a \cdot \varepsilon} - 1\right)} \]

      16. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \color{blue}{\left(e^{a \cdot \varepsilon} - 1\right)}} \]

      17. lift-exp.f64 N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \left(\color{blue}{e^{a \cdot \varepsilon}} - 1\right)} \]

      18. lower-expm1.f64 4.0

          \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}} \]

   3. Applied rewrites 4.0%

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\left(b + a\right) \cdot \varepsilon}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(b + a\right)}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\varepsilon \cdot \color{blue}{\left(b + a\right)}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b + \varepsilon \cdot a}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{b \cdot \varepsilon} + \varepsilon \cdot a\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{b \cdot \varepsilon} + \varepsilon \cdot a\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a + b \cdot \varepsilon}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      8. lift-fma.f64 4.0

         \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\varepsilon, a, b \cdot \varepsilon\right)}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\varepsilon, a, \color{blue}{b \cdot \varepsilon}\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\varepsilon, a, \color{blue}{\varepsilon \cdot b}\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

      11. lower-*.f64 4.0

          \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\varepsilon, a, \color{blue}{\varepsilon \cdot b}\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

   5. Applied rewrites 4.0%

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\varepsilon, a, \varepsilon \cdot b\right)}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{a}} \]
   7. Step-by-step derivation

      1. inv-pow N/A

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

      2. lower-pow.f64 77.8

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

   8. Applied rewrites 77.8%

      \[\leadsto \color{blue}{{a}^{-1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 60.7% accurate, 17.5× speedup

Math

\[\begin{array}{l} [a, b, eps] = \mathsf{sort}([a, b, eps])\\ \\ \frac{a + b}{a \cdot b} \end{array} \]

FPCore

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
(FPCore (a b eps) :precision binary64 (/ (+ a b) (* a b)))

C

assert(a < b && b < eps);
double code(double a, double b, double eps) {
	return (a + b) / (a * b);
}

Fortran

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
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, b, eps)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: eps
    code = (a + b) / (a * b)
end function

Java

assert a < b && b < eps;
public static double code(double a, double b, double eps) {
	return (a + b) / (a * b);
}

Python

[a, b, eps] = sort([a, b, eps])
def code(a, b, eps):
	return (a + b) / (a * b)

Julia

a, b, eps = sort([a, b, eps])
function code(a, b, eps)
	return Float64(Float64(a + b) / Float64(a * b))
end

MATLAB

a, b, eps = num2cell(sort([a, b, eps])){:}
function tmp = code(a, b, eps)
	tmp = (a + b) / (a * b);
end

Wolfram

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
code[a_, b_, eps_] := N[(N[(a + b), $MachinePrecision] / N[(a * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.0%

   \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 0.0

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

   4. lift--.f64 N/A

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

   5. lift-exp.f64 N/A

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

   6. lower-expm1.f64 0.4

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)} \cdot \varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\left(a + b\right)} \cdot \varepsilon\right) \cdot \varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

   8. +-commutative N/A

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\left(b + a\right)} \cdot \varepsilon\right) \cdot \varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

   9. lower-+.f64 0.4

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\left(b + a\right)} \cdot \varepsilon\right) \cdot \varepsilon}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}} \]

   11. *-commutative N/A

       \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\left(e^{b \cdot \varepsilon} - 1\right) \cdot \left(e^{a \cdot \varepsilon} - 1\right)}} \]

   12. lower-*.f64 0.4

       \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\left(e^{b \cdot \varepsilon} - 1\right) \cdot \left(e^{a \cdot \varepsilon} - 1\right)}} \]

   13. lift--.f64 N/A

       \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\left(e^{b \cdot \varepsilon} - 1\right)} \cdot \left(e^{a \cdot \varepsilon} - 1\right)} \]

   14. lift-exp.f64 N/A

       \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\left(\color{blue}{e^{b \cdot \varepsilon}} - 1\right) \cdot \left(e^{a \cdot \varepsilon} - 1\right)} \]

   15. lower-expm1.f64 0.3

       \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\color{blue}{\mathsf{expm1}\left(b \cdot \varepsilon\right)} \cdot \left(e^{a \cdot \varepsilon} - 1\right)} \]

   16. lift--.f64 N/A

       \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \color{blue}{\left(e^{a \cdot \varepsilon} - 1\right)}} \]

   17. lift-exp.f64 N/A

       \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \left(\color{blue}{e^{a \cdot \varepsilon}} - 1\right)} \]

   18. lower-expm1.f64 8.9

       \[\leadsto \frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \color{blue}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}} \]

3. Applied rewrites 8.9%

   \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)}} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\left(b + a\right) \cdot \varepsilon}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot \left(b + a\right)}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

   3. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{expm1}\left(\varepsilon \cdot \color{blue}{\left(b + a\right)}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

   4. distribute-lft-in N/A

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot b + \varepsilon \cdot a}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{b \cdot \varepsilon} + \varepsilon \cdot a\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{b \cdot \varepsilon} + \varepsilon \cdot a\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

   7. +-commutative N/A

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\varepsilon \cdot a + b \cdot \varepsilon}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

   8. lift-fma.f64 8.9

      \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\varepsilon, a, b \cdot \varepsilon\right)}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\varepsilon, a, \color{blue}{b \cdot \varepsilon}\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\varepsilon, a, \color{blue}{\varepsilon \cdot b}\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

   11. lower-*.f64 8.9

       \[\leadsto \frac{\mathsf{expm1}\left(\mathsf{fma}\left(\varepsilon, a, \color{blue}{\varepsilon \cdot b}\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

5. Applied rewrites 8.9%

   \[\leadsto \frac{\mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\varepsilon, a, \varepsilon \cdot b\right)}\right) \cdot \varepsilon}{\mathsf{expm1}\left(b \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(a \cdot \varepsilon\right)} \]

6. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\frac{a + b}{a \cdot b}} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{a + b}{\color{blue}{a \cdot b}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{a + b}{\color{blue}{a} \cdot b} \]

   3. lower-*.f64 60.7

      \[\leadsto \frac{a + b}{a \cdot \color{blue}{b}} \]

8. Applied rewrites 60.7%

   \[\leadsto \color{blue}{\frac{a + b}{a \cdot b}} \]
9. Add Preprocessing

Developer Target 1: 100.0% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{a} + \frac{1}{b} \end{array} \]

FPCore

(FPCore (a b eps) :precision binary64 (+ (/ 1.0 a) (/ 1.0 b)))

C

double code(double a, double b, double eps) {
	return (1.0 / a) + (1.0 / b);
}

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, b, eps)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: eps
    code = (1.0d0 / a) + (1.0d0 / b)
end function

Java

public static double code(double a, double b, double eps) {
	return (1.0 / a) + (1.0 / b);
}

Python

def code(a, b, eps):
	return (1.0 / a) + (1.0 / b)

Julia

function code(a, b, eps)
	return Float64(Float64(1.0 / a) + Float64(1.0 / b))
end

MATLAB

function tmp = code(a, b, eps)
	tmp = (1.0 / a) + (1.0 / b);
end

Wolfram

code[a_, b_, eps_] := N[(N[(1.0 / a), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025107 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :precision binary64
  :pre (and (and (<= (fabs a) 710.0) (<= (fabs b) 710.0)) (and (<= (* 1e-27 (fmin (fabs a) (fabs b))) eps) (<= eps (fmin (fabs a) (fabs b)))))

  :alt
  (! :herbie-platform default (+ (/ 1 a) (/ 1 b)))

  (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))