expq3 (problem 3.4.2)

Percentage Accurate: 6.4% → 96.7%

Time: 19.9s

Alternatives: 6

Speedup: 107.0×

• could not determine a ground truth

  1. a = -2.00000000000000007e154
  2. b = 1.9999999999999999e-154
  3. eps = -1.9999999999999999e-77

Specification

\[-1 < \varepsilon \land \varepsilon < 1\]

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: 6.4% 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: 96.7% accurate, 21.4× speedup

Math

\[\begin{array}{l} \\ \left(\left(0.5 \cdot \varepsilon + \frac{1}{b}\right) + \frac{1}{a}\right) - 0.5 \cdot \varepsilon \end{array} \]

FPCore

(FPCore (a b eps)
 :precision binary64
 (- (+ (+ (* 0.5 eps) (/ 1.0 b)) (/ 1.0 a)) (* 0.5 eps)))

C

double code(double a, double b, double eps) {
	return (((0.5 * eps) + (1.0 / b)) + (1.0 / a)) - (0.5 * eps);
}

Fortran

real(8) function code(a, b, eps)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: eps
    code = (((0.5d0 * eps) + (1.0d0 / b)) + (1.0d0 / a)) - (0.5d0 * eps)
end function

Java

public static double code(double a, double b, double eps) {
	return (((0.5 * eps) + (1.0 / b)) + (1.0 / a)) - (0.5 * eps);
}

Python

def code(a, b, eps):
	return (((0.5 * eps) + (1.0 / b)) + (1.0 / a)) - (0.5 * eps)

Julia

function code(a, b, eps)
	return Float64(Float64(Float64(Float64(0.5 * eps) + Float64(1.0 / b)) + Float64(1.0 / a)) - Float64(0.5 * eps))
end

MATLAB

function tmp = code(a, b, eps)
	tmp = (((0.5 * eps) + (1.0 / b)) + (1.0 / a)) - (0.5 * eps);
end

Wolfram

code[a_, b_, eps_] := N[(N[(N[(N[(0.5 * eps), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision] - N[(0.5 * eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.3%

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

   1. *-commutative 6.3%

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

   2. associate-*l/ 6.3%

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

   3. *-commutative 6.3%

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

   4. expm1-def 7.9%

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

   5. *-commutative 7.9%

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

   6. associate-/r* 7.9%

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

   7. expm1-def 18.5%

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

   8. *-commutative 18.5%

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

   9. expm1-def 47.7%

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

   10. *-commutative 47.7%

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

3. Simplified 47.7%

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

4. Taylor expanded in a around 0 15.0%

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

5. Taylor expanded in eps around 0 98.3%

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

6. Final simplification 98.3%

   \[\leadsto \left(\left(0.5 \cdot \varepsilon + \frac{1}{b}\right) + \frac{1}{a}\right) - 0.5 \cdot \varepsilon \]

Alternative 2: 95.1% accurate, 29.2× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{b} + \frac{1}{a}\right) - 0.5 \cdot \varepsilon \end{array} \]

FPCore

(FPCore (a b eps) :precision binary64 (- (+ (/ 1.0 b) (/ 1.0 a)) (* 0.5 eps)))

C

double code(double a, double b, double eps) {
	return ((1.0 / b) + (1.0 / a)) - (0.5 * eps);
}

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 / b) + (1.0d0 / a)) - (0.5d0 * eps)
end function

Java

public static double code(double a, double b, double eps) {
	return ((1.0 / b) + (1.0 / a)) - (0.5 * eps);
}

Python

def code(a, b, eps):
	return ((1.0 / b) + (1.0 / a)) - (0.5 * eps)

Julia

function code(a, b, eps)
	return Float64(Float64(Float64(1.0 / b) + Float64(1.0 / a)) - Float64(0.5 * eps))
end

MATLAB

function tmp = code(a, b, eps)
	tmp = ((1.0 / b) + (1.0 / a)) - (0.5 * eps);
end

Wolfram

code[a_, b_, eps_] := N[(N[(N[(1.0 / b), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision] - N[(0.5 * eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.3%

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

   1. *-commutative 6.3%

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

   2. times-frac 6.3%

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

   3. +-commutative 6.3%

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

   4. expm1-def 17.9%

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

   5. *-commutative 17.9%

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

   6. expm1-def 18.5%

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

   7. +-commutative 18.5%

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

   8. *-commutative 18.5%

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

   9. expm1-def 53.3%

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

   10. *-commutative 53.3%

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

3. Simplified 53.3%

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

4. Taylor expanded in eps around 0 54.7%

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

5. Taylor expanded in b around 0 95.1%

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

6. Final simplification 95.1%

   \[\leadsto \left(\frac{1}{b} + \frac{1}{a}\right) - 0.5 \cdot \varepsilon \]

Alternative 3: 94.7% accurate, 45.9× speedup

Math

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

FPCore

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

C

double code(double a, double b, double eps) {
	return (1.0 / b) + (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 / b) + (1.0d0 / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.3%

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

   1. *-commutative 6.3%

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

   2. associate-*l/ 6.3%

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

   3. *-commutative 6.3%

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

   4. expm1-def 7.9%

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

   5. *-commutative 7.9%

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

   6. associate-/r* 7.9%

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

   7. expm1-def 18.5%

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

   8. *-commutative 18.5%

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

   9. expm1-def 47.7%

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

   10. *-commutative 47.7%

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

3. Simplified 47.7%

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

4. Taylor expanded in eps around 0 78.5%

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

5. Taylor expanded in a around 0 95.0%

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

6. Final simplification 95.0%

   \[\leadsto \frac{1}{b} + \frac{1}{a} \]

Alternative 4: 59.3% accurate, 63.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-112}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b eps)
 :precision binary64
 (if (<= a -1.9e-112) (/ 1.0 b) (/ 1.0 a)))

C

double code(double a, double b, double eps) {
	double tmp;
	if (a <= -1.9e-112) {
		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 <= (-1.9d-112)) 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 <= -1.9e-112) {
		tmp = 1.0 / b;
	} else {
		tmp = 1.0 / a;
	}
	return tmp;
}

Python

def code(a, b, eps):
	tmp = 0
	if a <= -1.9e-112:
		tmp = 1.0 / b
	else:
		tmp = 1.0 / a
	return tmp

Julia

function code(a, b, eps)
	tmp = 0.0
	if (a <= -1.9e-112)
		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 <= -1.9e-112)
		tmp = 1.0 / b;
	else
		tmp = 1.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, eps_] := If[LessEqual[a, -1.9e-112], N[(1.0 / b), $MachinePrecision], N[(1.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.89999999999999997e-112

   1. Initial program 8.8%

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

      1. *-commutative 8.8%

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

      2. associate-*l/ 8.8%

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

      3. *-commutative 8.8%

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

      4. expm1-def 10.3%

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

      5. *-commutative 10.3%

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

      6. associate-/r* 10.3%

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

      7. expm1-def 28.1%

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

      8. *-commutative 28.1%

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

      9. expm1-def 69.3%

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

      10. *-commutative 69.3%

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

   3. Simplified 69.3%

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

   4. Taylor expanded in b around 0 66.1%

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

   if -1.89999999999999997e-112 < a

   1. Initial program 5.3%

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

      1. *-commutative 5.3%

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

      2. associate-*l/ 5.3%

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

      3. *-commutative 5.3%

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

      4. expm1-def 6.9%

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

      5. *-commutative 6.9%

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

      6. associate-/r* 6.9%

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

      7. expm1-def 14.5%

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

      8. *-commutative 14.5%

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

      9. expm1-def 38.5%

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

      10. *-commutative 38.5%

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

   3. Simplified 38.5%

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

   4. Taylor expanded in a around 0 58.4%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-112}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

Alternative 5: 3.2% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot -0.5 \end{array} \]

FPCore

(FPCore (a b eps) :precision binary64 (* eps -0.5))

C

double code(double a, double b, double eps) {
	return eps * -0.5;
}

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 * (-0.5d0)
end function

Java

public static double code(double a, double b, double eps) {
	return eps * -0.5;
}

Python

def code(a, b, eps):
	return eps * -0.5

Julia

function code(a, b, eps)
	return Float64(eps * -0.5)
end

MATLAB

function tmp = code(a, b, eps)
	tmp = eps * -0.5;
end

Wolfram

code[a_, b_, eps_] := N[(eps * -0.5), $MachinePrecision]

Derivation

1. Initial program 6.3%

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

   1. *-commutative 6.3%

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

   2. times-frac 6.3%

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

   3. +-commutative 6.3%

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

   4. expm1-def 17.9%

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

   5. *-commutative 17.9%

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

   6. expm1-def 18.5%

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

   7. +-commutative 18.5%

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

   8. *-commutative 18.5%

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

   9. expm1-def 53.3%

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

   10. *-commutative 53.3%

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

3. Simplified 53.3%

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

4. Taylor expanded in eps around 0 54.7%

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

5. Taylor expanded in b around 0 95.1%

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

6. Taylor expanded in eps around inf 2.9%

   \[\leadsto \color{blue}{-0.5 \cdot \varepsilon} \]
7. Step-by-step derivation

   1. *-commutative 2.9%

      \[\leadsto \color{blue}{\varepsilon \cdot -0.5} \]

8. Simplified 2.9%

   \[\leadsto \color{blue}{\varepsilon \cdot -0.5} \]

9. Final simplification 2.9%

   \[\leadsto \varepsilon \cdot -0.5 \]

Alternative 6: 47.9% 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

1. Initial program 6.3%

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

   1. *-commutative 6.3%

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

   2. associate-*l/ 6.3%

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

   3. *-commutative 6.3%

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

   4. expm1-def 7.9%

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

   5. *-commutative 7.9%

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

   6. associate-/r* 7.9%

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

   7. expm1-def 18.5%

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

   8. *-commutative 18.5%

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

   9. expm1-def 47.7%

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

   10. *-commutative 47.7%

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

3. Simplified 47.7%

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

4. Taylor expanded in a around 0 49.3%

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

5. Final simplification 49.3%

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

Developer target: 77.8% accurate, 45.9× speedup

Math

\[\begin{array}{l} \\ \frac{a + b}{a \cdot b} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023274 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :precision binary64
  :pre (and (< -1.0 eps) (< eps 1.0))

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

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