expq3 (problem 3.4.2)

Percentage Accurate: 0.0% → 100.0%

Time: 23.1s

Alternatives: 4

Speedup: 107.0×

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: 100.0% accurate, 45.9× 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]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 2: 59.0% accurate, 24.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.9 \cdot 10^{-202}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{elif}\;a \leq -1 \cdot 10^{-210} \lor \neg \left(a \leq -1.15 \cdot 10^{-227}\right):\\ \;\;\;\;\frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{a}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b eps)
 :precision binary64
 (if (<= a -6.9e-202)
   (/ 1.0 b)
   (if (or (<= a -1e-210) (not (<= a -1.15e-227)))
     (/ 1.0 a)
     (* (/ 1.0 a) (/ a b)))))

C

double code(double a, double b, double eps) {
	double tmp;
	if (a <= -6.9e-202) {
		tmp = 1.0 / b;
	} else if ((a <= -1e-210) || !(a <= -1.15e-227)) {
		tmp = 1.0 / a;
	} else {
		tmp = (1.0 / a) * (a / b);
	}
	return tmp;
}

Fortran

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 (a <= (-6.9d-202)) then
        tmp = 1.0d0 / b
    else if ((a <= (-1d-210)) .or. (.not. (a <= (-1.15d-227)))) then
        tmp = 1.0d0 / a
    else
        tmp = (1.0d0 / a) * (a / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double eps) {
	double tmp;
	if (a <= -6.9e-202) {
		tmp = 1.0 / b;
	} else if ((a <= -1e-210) || !(a <= -1.15e-227)) {
		tmp = 1.0 / a;
	} else {
		tmp = (1.0 / a) * (a / b);
	}
	return tmp;
}

Python

def code(a, b, eps):
	tmp = 0
	if a <= -6.9e-202:
		tmp = 1.0 / b
	elif (a <= -1e-210) or not (a <= -1.15e-227):
		tmp = 1.0 / a
	else:
		tmp = (1.0 / a) * (a / b)
	return tmp

Julia

function code(a, b, eps)
	tmp = 0.0
	if (a <= -6.9e-202)
		tmp = Float64(1.0 / b);
	elseif ((a <= -1e-210) || !(a <= -1.15e-227))
		tmp = Float64(1.0 / a);
	else
		tmp = Float64(Float64(1.0 / a) * Float64(a / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, eps)
	tmp = 0.0;
	if (a <= -6.9e-202)
		tmp = 1.0 / b;
	elseif ((a <= -1e-210) || ~((a <= -1.15e-227)))
		tmp = 1.0 / a;
	else
		tmp = (1.0 / a) * (a / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, eps_] := If[LessEqual[a, -6.9e-202], N[(1.0 / b), $MachinePrecision], If[Or[LessEqual[a, -1e-210], N[Not[LessEqual[a, -1.15e-227]], $MachinePrecision]], N[(1.0 / a), $MachinePrecision], N[(N[(1.0 / a), $MachinePrecision] * N[(a / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 3: 59.0% accurate, 34.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.7 \cdot 10^{-202} \lor \neg \left(a \leq -5.2 \cdot 10^{-212}\right) \land a \leq -1.15 \cdot 10^{-227}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b eps)
 :precision binary64
 (if (or (<= a -5.7e-202) (and (not (<= a -5.2e-212)) (<= a -1.15e-227)))
   (/ 1.0 b)
   (/ 1.0 a)))

C

double code(double a, double b, double eps) {
	double tmp;
	if ((a <= -5.7e-202) || (!(a <= -5.2e-212) && (a <= -1.15e-227))) {
		tmp = 1.0 / b;
	} else {
		tmp = 1.0 / a;
	}
	return tmp;
}

Fortran

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 ((a <= (-5.7d-202)) .or. (.not. (a <= (-5.2d-212))) .and. (a <= (-1.15d-227))) then
        tmp = 1.0d0 / b
    else
        tmp = 1.0d0 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double eps) {
	double tmp;
	if ((a <= -5.7e-202) || (!(a <= -5.2e-212) && (a <= -1.15e-227))) {
		tmp = 1.0 / b;
	} else {
		tmp = 1.0 / a;
	}
	return tmp;
}

Python

def code(a, b, eps):
	tmp = 0
	if (a <= -5.7e-202) or (not (a <= -5.2e-212) and (a <= -1.15e-227)):
		tmp = 1.0 / b
	else:
		tmp = 1.0 / a
	return tmp

Julia

function code(a, b, eps)
	tmp = 0.0
	if ((a <= -5.7e-202) || (!(a <= -5.2e-212) && (a <= -1.15e-227)))
		tmp = Float64(1.0 / b);
	else
		tmp = Float64(1.0 / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, eps)
	tmp = 0.0;
	if ((a <= -5.7e-202) || (~((a <= -5.2e-212)) && (a <= -1.15e-227)))
		tmp = 1.0 / b;
	else
		tmp = 1.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, eps_] := If[Or[LessEqual[a, -5.7e-202], And[N[Not[LessEqual[a, -5.2e-212]], $MachinePrecision], LessEqual[a, -1.15e-227]]], N[(1.0 / b), $MachinePrecision], N[(1.0 / a), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 4: 49.6% accurate, 107.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Developer target: 100.0% accurate, 45.9× 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 2023350 
(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)))))

  :herbie-target
  (+ (/ 1.0 a) (/ 1.0 b))

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