expq3 (problem 3.4.2)

Average Error: 60.3 → 3.1

Time: 17.0s

Precision: binary64

Cost: 960

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

Math

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

↓

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

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
 (+ (* -0.5 (/ (* eps b) b)) (+ (/ 1.0 a) (/ 1.0 b))))

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 (-0.5 * ((eps * b) / b)) + ((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 = (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 = ((-0.5d0) * ((eps * b) / b)) + ((1.0d0 / a) + (1.0d0 / b))
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 (-0.5 * ((eps * b) / b)) + ((1.0 / a) + (1.0 / b));
}

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 (-0.5 * ((eps * b) / b)) + ((1.0 / a) + (1.0 / b))

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(-0.5 * Float64(Float64(eps * b) / b)) + Float64(Float64(1.0 / a) + Float64(1.0 / b)))
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 = (-0.5 * ((eps * b) / b)) + ((1.0 / a) + (1.0 / b));
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[(-0.5 * N[(N[(eps * b), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / a), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	60.3
Target	14.4
Herbie	3.1

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

Derivation

1. Initial program 60.3

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

2. Simplified 30.6

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

3. Taylor expanded in eps around 0 38.2

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

4. Taylor expanded in eps around 0 39.2

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

5. Simplified 39.2

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

6. Taylor expanded in eps around 0 10.4

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

7. Taylor expanded in a around 0 3.1

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

8. Final simplification 3.1

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

Alternatives

Alternative 1
Error	26.9
Cost	844

\[\begin{array}{l} \mathbf{if}\;b \leq 4.783125571545917 \cdot 10^{-235}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{elif}\;b \leq 1.389625510999873 \cdot 10^{-223}:\\ \;\;\;\;\frac{1}{a}\\ \mathbf{elif}\;b \leq 2.799648529661814 \cdot 10^{-146}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + -0.5 \cdot \varepsilon\\ \end{array} \]

Alternative 2
Error	26.8
Cost	844

\[\begin{array}{l} t_0 := \frac{1}{b} + -0.5 \cdot \varepsilon\\ \mathbf{if}\;b \leq 4.783125571545917 \cdot 10^{-235}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.389625510999873 \cdot 10^{-223}:\\ \;\;\;\;\frac{1}{a}\\ \mathbf{elif}\;b \leq 2.799648529661814 \cdot 10^{-146}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + -0.5 \cdot \varepsilon\\ \end{array} \]

Alternative 3
Error	27.0
Cost	588

\[\begin{array}{l} \mathbf{if}\;b \leq 4.783125571545917 \cdot 10^{-235}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{elif}\;b \leq 1.389625510999873 \cdot 10^{-223}:\\ \;\;\;\;\frac{1}{a}\\ \mathbf{elif}\;b \leq 2.799648529661814 \cdot 10^{-146}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array} \]

Alternative 4
Error	3.4
Cost	448

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

Alternative 5
Error	62.0
Cost	192

\[-0.5 \cdot \varepsilon \]

Alternative 6
Error	33.3
Cost	192

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

Reproduce

herbie shell --seed 2022228 
(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))))