expq3 (problem 3.4.2)

Percentage Accurate: 0.0% → 100.0%

Time: 26.6s

Alternatives: 4

Speedup: 349.0×

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, 3.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double eps) {
	return pow(b, -1.0) - (-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 = (b ** (-1.0d0)) - ((-1.0d0) / a)
end function

Java

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

Python

def code(a, b, eps):
	return math.pow(b, -1.0) - (-1.0 / a)

Julia

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

MATLAB

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

Wolfram

code[a_, b_, eps_] := N[(N[Power[b, -1.0], $MachinePrecision] - N[(-1.0 / a), $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. Add Preprocessing

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 99.8

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in a around inf

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

8. Applied rewrites 99.9%

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

9. Final simplification 99.9%

   \[\leadsto {b}^{-1} - \frac{-1}{a} \]
10. Add Preprocessing

Alternative 2: 60.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{-188}:\\ \;\;\;\;{b}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b eps)
 :precision binary64
 (if (<= b 2.2e-188) (pow b -1.0) (pow a -1.0)))

C

double code(double a, double b, double eps) {
	double tmp;
	if (b <= 2.2e-188) {
		tmp = pow(b, -1.0);
	} else {
		tmp = pow(a, -1.0);
	}
	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 (b <= 2.2d-188) then
        tmp = b ** (-1.0d0)
    else
        tmp = a ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double eps) {
	double tmp;
	if (b <= 2.2e-188) {
		tmp = Math.pow(b, -1.0);
	} else {
		tmp = Math.pow(a, -1.0);
	}
	return tmp;
}

Python

def code(a, b, eps):
	tmp = 0
	if b <= 2.2e-188:
		tmp = math.pow(b, -1.0)
	else:
		tmp = math.pow(a, -1.0)
	return tmp

Julia

function code(a, b, eps)
	tmp = 0.0
	if (b <= 2.2e-188)
		tmp = b ^ -1.0;
	else
		tmp = a ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, eps)
	tmp = 0.0;
	if (b <= 2.2e-188)
		tmp = b ^ -1.0;
	else
		tmp = a ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, eps_] := If[LessEqual[b, 2.2e-188], N[Power[b, -1.0], $MachinePrecision], N[Power[a, -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.2e-188

   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 55.8

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

   5. Applied rewrites 55.8%

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

   if 2.2e-188 < 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 76.7

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

   5. Applied rewrites 76.7%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{-188}:\\ \;\;\;\;{b}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{a}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ {a}^{-1} \end{array} \]

FPCore

(FPCore (a b eps) :precision binary64 (pow a -1.0))

C

double code(double a, double b, double eps) {
	return pow(a, -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 = a ** (-1.0d0)
end function

Java

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

Python

def code(a, b, eps):
	return math.pow(a, -1.0)

Julia

function code(a, b, eps)
	return a ^ -1.0
end

MATLAB

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

Wolfram

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 51.9

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

5. Applied rewrites 51.9%

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

6. Final simplification 51.9%

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

Alternative 4: 2.0% accurate, 349.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b eps) :precision binary64 0.0)

C

double code(double a, double b, double eps) {
	return 0.0;
}

Fortran

real(8) function code(a, b, eps)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double a, double b, double eps) {
	return 0.0;
}

Python

def code(a, b, eps):
	return 0.0

Julia

function code(a, b, eps)
	return 0.0
end

MATLAB

function tmp = code(a, b, eps)
	tmp = 0.0;
end

Wolfram

code[a_, b_, eps_] := 0.0

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 eps around 0

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

5. Applied rewrites 17.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(b \cdot a\right) \cdot \left(b + a\right)\right) \cdot -0.5, \frac{b + a}{\left(\left(b \cdot a\right) \cdot a\right) \cdot b}, \frac{{\left(b + a\right)}^{2}}{b} \cdot \frac{0.5}{a}\right), \varepsilon, \frac{1}{b} + \frac{1}{a}\right)} \]

6. Taylor expanded in eps around inf

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

8. Applied rewrites 2.0%

   \[\leadsto 0 \]
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

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 2024324 
(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))))