expq3 (problem 3.4.2)

Percentage Accurate: 0.0% → 99.8%

Time: 25.5s

Alternatives: 3

Speedup: 13.4×

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

real(8) function code(a, b, eps)
    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

real(8) function code(a, b, eps)
    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: 99.8% accurate, 13.4× speedup

Math

\[\begin{array}{l} [a, b, eps] = \mathsf{sort}([a, b, eps])\\ \\ \frac{\frac{b}{a} + 1}{b} \end{array} \]

FPCore

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
(FPCore (a b eps) :precision binary64 (/ (+ (/ b a) 1.0) b))

C

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

Fortran

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
real(8) function code(a, b, eps)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: eps
    code = ((b / a) + 1.0d0) / b
end function

Java

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

Python

[a, b, eps] = sort([a, b, eps])
def code(a, b, eps):
	return ((b / a) + 1.0) / b

Julia

a, b, eps = sort([a, b, eps])
function code(a, b, eps)
	return Float64(Float64(Float64(b / a) + 1.0) / b)
end

MATLAB

a, b, eps = num2cell(sort([a, b, eps])){:}
function tmp = code(a, b, eps)
	tmp = ((b / a) + 1.0) / b;
end

Wolfram

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
code[a_, b_, eps_] := N[(N[(N[(b / a), $MachinePrecision] + 1.0), $MachinePrecision] / b), $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. Add Preprocessing

3. Taylor expanded in b around 0

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 43.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\varepsilon \cdot \left(-0.5 + \frac{{\left(e^{a}\right)}^{\varepsilon}}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}\right), b, 1\right)}{b}} \]

6. Taylor expanded in eps around 0

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

8. Applied rewrites 99.7%

   \[\leadsto \frac{\frac{b}{a} + 1}{b} \]
9. Add Preprocessing

Alternative 2: 80.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} [a, b, eps] = \mathsf{sort}([a, b, eps])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{-171}:\\ \;\;\;\;{b}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-1}\\ \end{array} \end{array} \]

FPCore

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
(FPCore (a b eps)
 :precision binary64
 (if (<= b 1.05e-171) (pow b -1.0) (pow a -1.0)))

C

assert(a < b && b < eps);
double code(double a, double b, double eps) {
	double tmp;
	if (b <= 1.05e-171) {
		tmp = pow(b, -1.0);
	} else {
		tmp = pow(a, -1.0);
	}
	return tmp;
}

Fortran

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
real(8) function code(a, b, eps)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (b <= 1.05d-171) then
        tmp = b ** (-1.0d0)
    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 (b <= 1.05e-171) {
		tmp = Math.pow(b, -1.0);
	} 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 b <= 1.05e-171:
		tmp = math.pow(b, -1.0)
	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 (b <= 1.05e-171)
		tmp = b ^ -1.0;
	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 (b <= 1.05e-171)
		tmp = b ^ -1.0;
	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[b, 1.05e-171], N[Power[b, -1.0], $MachinePrecision], N[Power[a, -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.05e-171

   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. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 56.1

         \[\leadsto \color{blue}{\frac{1}{b}} \]

   5. Applied rewrites 56.1%

      \[\leadsto \color{blue}{\frac{1}{b}} \]

   if 1.05e-171 < b

   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. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 72.5

         \[\leadsto \color{blue}{\frac{1}{a}} \]

   5. Applied rewrites 72.5%

      \[\leadsto \color{blue}{\frac{1}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{-171}:\\ \;\;\;\;{b}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} [a, b, eps] = \mathsf{sort}([a, b, eps])\\ \\ {a}^{-1} \end{array} \]

FPCore

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
(FPCore (a b eps) :precision binary64 (pow a -1.0))

C

assert(a < b && b < eps);
double code(double a, double b, double eps) {
	return pow(a, -1.0);
}

Fortran

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
real(8) function code(a, b, eps)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: eps
    code = a ** (-1.0d0)
end function

Java

assert a < b && b < eps;
public static double code(double a, double b, double eps) {
	return Math.pow(a, -1.0);
}

Python

[a, b, eps] = sort([a, b, eps])
def code(a, b, eps):
	return math.pow(a, -1.0)

Julia

a, b, eps = sort([a, b, eps])
function code(a, b, eps)
	return a ^ -1.0
end

MATLAB

a, b, eps = num2cell(sort([a, b, eps])){:}
function tmp = code(a, b, eps)
	tmp = a ^ -1.0;
end

Wolfram

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
code[a_, b_, eps_] := N[Power[a, -1.0], $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. Add Preprocessing

3. Taylor expanded in a around 0

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

   1. lower-/.f64 52.4

      \[\leadsto \color{blue}{\frac{1}{a}} \]

5. Applied rewrites 52.4%

   \[\leadsto \color{blue}{\frac{1}{a}} \]

6. Final simplification 52.4%

   \[\leadsto {a}^{-1} \]
7. Add Preprocessing

Developer Target 1: 99.9% 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

real(8) function code(a, b, eps)
    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 2024314 
(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))))