expq3 (problem 3.4.2)

Percentage Accurate: 0.0% → 99.8%

Time: 26.1s

Alternatives: 4

Speedup: 29.1×

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 45.3%

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

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: 80.0% accurate, 10.9× speedup

Math

\[\begin{array}{l} [a, b, eps] = \mathsf{sort}([a, b, eps])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-196}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-140}:\\ \;\;\;\;\frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a}{b \cdot a}\\ \end{array} \end{array} \]

FPCore

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
(FPCore (a b eps)
 :precision binary64
 (if (<= b 2.1e-196)
   (/ 1.0 b)
   (if (<= b 2.5e-140) (/ 1.0 a) (/ (+ b a) (* b a)))))

C

assert(a < b && b < eps);
double code(double a, double b, double eps) {
	double tmp;
	if (b <= 2.1e-196) {
		tmp = 1.0 / b;
	} else if (b <= 2.5e-140) {
		tmp = 1.0 / a;
	} else {
		tmp = (b + a) / (b * a);
	}
	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 <= 2.1d-196) then
        tmp = 1.0d0 / b
    else if (b <= 2.5d-140) then
        tmp = 1.0d0 / a
    else
        tmp = (b + a) / (b * a)
    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 <= 2.1e-196) {
		tmp = 1.0 / b;
	} else if (b <= 2.5e-140) {
		tmp = 1.0 / a;
	} else {
		tmp = (b + a) / (b * a);
	}
	return tmp;
}

Python

[a, b, eps] = sort([a, b, eps])
def code(a, b, eps):
	tmp = 0
	if b <= 2.1e-196:
		tmp = 1.0 / b
	elif b <= 2.5e-140:
		tmp = 1.0 / a
	else:
		tmp = (b + a) / (b * a)
	return tmp

Julia

a, b, eps = sort([a, b, eps])
function code(a, b, eps)
	tmp = 0.0
	if (b <= 2.1e-196)
		tmp = Float64(1.0 / b);
	elseif (b <= 2.5e-140)
		tmp = Float64(1.0 / a);
	else
		tmp = Float64(Float64(b + a) / Float64(b * a));
	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 <= 2.1e-196)
		tmp = 1.0 / b;
	elseif (b <= 2.5e-140)
		tmp = 1.0 / a;
	else
		tmp = (b + a) / (b * a);
	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, 2.1e-196], N[(1.0 / b), $MachinePrecision], If[LessEqual[b, 2.5e-140], N[(1.0 / a), $MachinePrecision], N[(N[(b + a), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.09999999999999988e-196

   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 61.1

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

   5. Applied rewrites 61.1%

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

   if 2.09999999999999988e-196 < b < 2.50000000000000007e-140

   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 47.3

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

   5. Applied rewrites 47.3%

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

   if 2.50000000000000007e-140 < 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 eps around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 86.1

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

   5. Applied rewrites 86.1%

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

Alternative 3: 80.7% accurate, 19.4× speedup

Math

\[\begin{array}{l} [a, b, eps] = \mathsf{sort}([a, b, eps])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-196}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \end{array} \]

FPCore

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
(FPCore (a b eps) :precision binary64 (if (<= b 2.1e-196) (/ 1.0 b) (/ 1.0 a)))

C

assert(a < b && b < eps);
double code(double a, double b, double eps) {
	double tmp;
	if (b <= 2.1e-196) {
		tmp = 1.0 / b;
	} else {
		tmp = 1.0 / a;
	}
	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 <= 2.1d-196) then
        tmp = 1.0d0 / b
    else
        tmp = 1.0d0 / a
    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 <= 2.1e-196) {
		tmp = 1.0 / b;
	} else {
		tmp = 1.0 / a;
	}
	return tmp;
}

Python

[a, b, eps] = sort([a, b, eps])
def code(a, b, eps):
	tmp = 0
	if b <= 2.1e-196:
		tmp = 1.0 / b
	else:
		tmp = 1.0 / a
	return tmp

Julia

a, b, eps = sort([a, b, eps])
function code(a, b, eps)
	tmp = 0.0
	if (b <= 2.1e-196)
		tmp = Float64(1.0 / b);
	else
		tmp = Float64(1.0 / a);
	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 <= 2.1e-196)
		tmp = 1.0 / b;
	else
		tmp = 1.0 / a;
	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, 2.1e-196], N[(1.0 / b), $MachinePrecision], N[(1.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.09999999999999988e-196

   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 61.1

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

   5. Applied rewrites 61.1%

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

   if 2.09999999999999988e-196 < 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 64.8

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

   5. Applied rewrites 64.8%

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

Alternative 4: 48.9% accurate, 29.1× speedup

Math

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

FPCore

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

C

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

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 = 1.0d0 / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: a, b, and eps should be sorted in increasing order before calling this function.
code[a_, b_, eps_] := N[(1.0 / a), $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 46.7

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

5. Applied rewrites 46.7%

   \[\leadsto \color{blue}{\frac{1}{a}} \]
6. 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

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 2024235 
(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))))