expq3 (problem 3.4.2)

Percentage Accurate: 0.0% → 99.3%

Time: 27.8s

Alternatives: 3

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: 99.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.3333333333333333}{a} \cdot \left(3 \cdot \frac{a + b}{b}\right) \end{array} \]

FPCore

(FPCore (a b eps)
 :precision binary64
 (* (/ 0.3333333333333333 a) (* 3.0 (/ (+ a b) b))))

C

double code(double a, double b, double eps) {
	return (0.3333333333333333 / a) * (3.0 * ((a + b) / 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 = (0.3333333333333333d0 / a) * (3.0d0 * ((a + b) / b))
end function

Java

public static double code(double a, double b, double eps) {
	return (0.3333333333333333 / a) * (3.0 * ((a + b) / b));
}

Python

def code(a, b, eps):
	return (0.3333333333333333 / a) * (3.0 * ((a + b) / b))

Julia

function code(a, b, eps)
	return Float64(Float64(0.3333333333333333 / a) * Float64(3.0 * Float64(Float64(a + b) / b)))
end

MATLAB

function tmp = code(a, b, eps)
	tmp = (0.3333333333333333 / a) * (3.0 * ((a + b) / b));
end

Wolfram

code[a_, b_, eps_] := N[(N[(0.3333333333333333 / a), $MachinePrecision] * N[(3.0 * N[(N[(a + b), $MachinePrecision] / b), $MachinePrecision]), $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

4. Applied rewrites 26.9%

   \[\leadsto \color{blue}{\frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \left(\varepsilon \cdot 3\right)\right)} \cdot \left(\left(1 + \left(e^{a \cdot \varepsilon} + e^{a \cdot \left(\varepsilon + \varepsilon\right)}\right)\right) \cdot \frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}\right)} \]

5. Taylor expanded in eps around 0

   \[\leadsto \frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \left(\varepsilon \cdot 3\right)\right)} \cdot \color{blue}{\left(3 \cdot \frac{a + b}{b}\right)} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \left(\varepsilon \cdot 3\right)\right)} \cdot \color{blue}{\left(3 \cdot \frac{a + b}{b}\right)} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \left(\varepsilon \cdot 3\right)\right)} \cdot \left(3 \cdot \color{blue}{\frac{a + b}{b}}\right) \]

   3. lower-+.f64 42.0

      \[\leadsto \frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \left(\varepsilon \cdot 3\right)\right)} \cdot \left(3 \cdot \frac{\color{blue}{a + b}}{b}\right) \]

7. Applied rewrites 42.0%

   \[\leadsto \frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \left(\varepsilon \cdot 3\right)\right)} \cdot \color{blue}{\left(3 \cdot \frac{a + b}{b}\right)} \]

8. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{3}}{a}} \cdot \left(3 \cdot \frac{a + b}{b}\right) \]
9. Step-by-step derivation

   1. lower-/.f64 99.3

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{a}} \cdot \left(3 \cdot \frac{a + b}{b}\right) \]

10. Applied rewrites 99.3%

    \[\leadsto \color{blue}{\frac{0.3333333333333333}{a}} \cdot \left(3 \cdot \frac{a + b}{b}\right) \]
11. Add Preprocessing

Alternative 2: 61.2% accurate, 19.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-184}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b eps) :precision binary64 (if (<= a -5e-184) (/ 1.0 b) (/ 1.0 a)))

C

double code(double a, double b, double eps) {
	double tmp;
	if (a <= -5e-184) {
		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 <= (-5d-184)) 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 <= -5e-184) {
		tmp = 1.0 / b;
	} else {
		tmp = 1.0 / a;
	}
	return tmp;
}

Python

def code(a, b, eps):
	tmp = 0
	if a <= -5e-184:
		tmp = 1.0 / b
	else:
		tmp = 1.0 / a
	return tmp

Julia

function code(a, b, eps)
	tmp = 0.0
	if (a <= -5e-184)
		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 <= -5e-184)
		tmp = 1.0 / b;
	else
		tmp = 1.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, eps_] := If[LessEqual[a, -5e-184], N[(1.0 / b), $MachinePrecision], N[(1.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.00000000000000003e-184

   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 76.1

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

   5. Applied rewrites 76.1%

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

   if -5.00000000000000003e-184 < 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. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 57.5

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

   5. Applied rewrites 57.5%

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

Alternative 3: 49.9% accurate, 29.1× 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

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 48.0

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

5. Applied rewrites 48.0%

   \[\leadsto \color{blue}{\frac{1}{a}} \]
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 2024212 
(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))))