expq3 (problem 3.4.2)

Average Error: 60.4 → 1.9

Time: 34.8s

Precision: binary64

Cost: 960

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

Math

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

↓

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

FPCore

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

↓

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

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));
}

↓

double code(double a, double b, double eps) {
	return (((1.0 / a) + (-0.5 * eps)) + (1.0 / b)) - (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 = (eps * (exp(((a + b) * eps)) - 1.0d0)) / ((exp((a * eps)) - 1.0d0) * (exp((b * eps)) - 1.0d0))
end 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) + ((-0.5d0) * eps)) + (1.0d0 / b)) - (0.5d0 * eps)
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));
}

↓

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

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))

↓

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

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

↓

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

↓

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

↓

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

Target

Original	60.4
Target	14.7
Herbie	1.9

\[\frac{a + b}{a \cdot b} \]

Derivation

1. Initial program 60.4

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

2. Taylor expanded in b around 0 55.4

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

3. Taylor expanded in eps around 0 55.2

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

4. Taylor expanded in eps around 0 1.9

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

5. Simplified 1.9

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

   [Start] 1.9	\[ \left(\left(-0.5 \cdot \varepsilon + \frac{1}{a}\right) + \frac{1}{b}\right) - 0.5 \cdot \varepsilon \]
   rational.json-simplify-1 [=>] 1.9	\[ \left(\color{blue}{\left(\frac{1}{a} + -0.5 \cdot \varepsilon\right)} + \frac{1}{b}\right) - 0.5 \cdot \varepsilon \]

6. Final simplification 1.9

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

Alternatives

Alternative 1
Error	3.0
Cost	704

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

Alternative 2
Error	3.3
Cost	448

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

Alternative 3
Error	26.2
Cost	324

\[\begin{array}{l} \mathbf{if}\;a \leq -1.62 \cdot 10^{-99}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

Alternative 4
Error	33.4
Cost	192

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

Alternative 5
Error	60.6
Cost	128

\[-\varepsilon \]

Reproduce

herbie shell --seed 2023077 
(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))))